I'm not convinced that your expression "all the strawberries he does have" is a recognized way of disambiguating the expression that you say is ambiguous: "all the strawberries he has." When would we use the expression "all the strawberries he does have"? As far as I can see, only in special contexts such as this one: "He doesn't have all the strawberries in the county. But all the strawberries he does have are organic." In that example, "does" isn't used to signal the indicative mood; instead it's used merely to emphasize a contrast.
Nor am I convinced that "does" + infinitive always carries existential import (i.e., implies the existence of at least one thing satisfying the verb phrase). Consider:
(P) "All the intelligent extraterrestrials our galaxy does contain are extraterrestrials."
Again, P will sound awkward except in a context such as this:
(Q) "Our galaxy may not contain any intelligent extraterrestrials. But all the intelligent extraterrestrials our galaxy does contain are extraterrestrials."
Whether or not you believe our galaxy contains intelligent extraterrestrials, it would be wrong to deny the second sentence in Q, wouldn't it?
Just what is a definition?
To answer your first question, I looked up "definition" (in the linguistic sense of the word) and got this: "define: to explain the meaning of (a word, phrase, etc.)." If that definition is accurate, then a definition is an explanation of the meaning of a word, phrase, etc.
Are definitions ever proved?
The definitions in dictionaries are attempts to explain the actual meanings of terms as those terms are used by the community of language-users. As such, definitions can be more accurate or less accurate, depending on how well they capture the actual way terms are used. I wouldn't say that such definitions are ever "proved," but as a matter of empirical fact some definitions are more accurate than others.
Another kind of definition, not found in dictionaries, is a stipulative definition: it's just a speaker's proposal to use a word in a particular way or else the speaker's declaration that he/she will be using the word in that way. Stipulative definitions aren't meant to report the actual use of the word by the community of language-users; instead, they're proposed or declared in order to have a shorthand way of saying something.
Are definitions all man-made?
Yes, in the sense that all definitions are invented by language-users rather than arising independently of language-users.
If they are man-made, what good are they?
As far as I know, all explanations (on Earth, anyway) are made by humans. Being explanations, dictionary definitions are useful in the way that explanations in general are useful. Stipulative definitions are useful as shorthand ways of saying something.
I would question your friend's claim that "the very first noodles ever made and the noodles most commonly eaten around the world are made from wheat by definition." I don't see the justification for the final two words in that claim. Even if the first noodles happen to have been made from wheat, I don't see how being made from wheat becomes part of the definition of 'noodle'. The first boats weren't made of fiberglass, but surely that doesn't preclude the existence of fiberglass boats or make 'fiberglass boat' a contradiction in terms.
It's a good idea to consult the SEP for discussion of these questions and for citations to various published answers. Continue to do so. I'd question, however, whether "most philosophers today" reject the analytic/synthetic distinction. According to the recent PhilPapers survey, 64.9% of "target faculty" either "accept or lean toward" accepting the distinction (see this link). Reports of its demise would appear to be exaggerated.
I strongly doubt that all definitions are human-made. Given the staggering number of stars that astronomers say exist, it seems highly likely that intelligent life has arisen in at least one other place -- intelligent life capable of creating languages and capable of creating explicit definitions for at least some of the items in those languages. All the definitions we currently know of are human-made, but the region of spacetime we've sampled is exceedingly small compared to what's out there.
The most obvious reason why counterfactual talk is taken seriously by philosophers is that it's virtually impossible to avoid it. We constantly find ourselves asking -- for good reason -- what would happen in certain circumstances, and so understanding more deeply what that sort of talk might amount to seems to be a reasonable project.
You offer a dilemma. We consider a counterfactual "If A were the case, then C would be the case." You then give us a choice between determinism and indeterminism. So suppose determinism is true. Then even if 'A' is false as things are, the deterministic story you're imagining can still be applied in a hypothetical case in which A is true. After all, we do that sort of thing all the time when we solve physics problems! If the result of applying the theory is that C also turns out to be true, then it's true as things actually are that if 'A' were true, 'C' would be true as well. Why is that vacuous? It's certainly not trivial; otherwise physics itself would be trivial.
On the other hand, if assuming 'A' rules out 'C,' then it's false as things actually are that if 'A' were true, 'C' would also be true. That's not vacuous, and it doesn't make the counterfactual a contradiction. Keep in mind: the laws of nature are contingent truths.
But in fact, we're over-simplifying. Suppose the world is deterministic. Suppose Johnny is about to strike a match. Will it light? Our two assumptions don't answer the question. Whether the match would light depends not just on the laws but also on the background conditions, as you're aware. But notice: when I say "If Johnny were to strike a match, it would light" I'm saying (on a Lewis/Stalknaker-type account) that in the non-actual situations that most closely resemble how things actually are except that Johnny strikes a match, the match lights. That's something I could well be wrong about, or right about, consistently with determinism and with the actual laws of the world. Whether Johnny's match lights in the nearest possible situations where he strikes isn't just obvious. It depends (among other things) on all sorts of contingent facts about the actual world, and these are facts about which I might well be mistaken. The point of spelling out truth conditions is to give an account of what being right or wrong would amount to.
Things don't change if we consider indeterministic worlds. One reason is that even if things aren't fully deterministic, there would still be true counterfactuals. Some aren't so interesting. For example: if I were 6' tall, I'd be over 5' 10." That's true, even if it's true as a matter of logic/mathematics. Others wouldn't have to be so trivial. There could be cases where strict causal relations hold even if not all events have strict causes. But suppose everything is, so to speak, loose and separate. Then it might be that all counterfactuals that aren't true as a matter of logic or math are false. (False, by the way; not indeterminate.) That would be a big deal but it wouldn't make the non-logical counterfactuals vacuous and it wouldn't make then contradictory. It would just make them false. It would also leave us with a lot of true "might"-counterfactuals. For example: if Johnny were to strike the match, it might light, and it might not. Lewis's account spells out truth conditions for "might" counterfactuals, and also allows us to state truth conditions for "would" counterfactuals in terms of "might." From "If it were the case that A then it might be the case that not-C," it follows on Lewis's account that "If it were the case that A then it would be the case that C" is false.
As for rigor: the everyday use of counterfactuals may lack rigor in various sorts of ways, but this isn't as bad as it might sound. The everyday use of language in general lacks various sorts of rigor, but that doesn't make the study of semantics pointless. And it's also worth keeping in mind: Lewis saw it as a virtue of his theory that it can take straightforward account of certain kinds of lack of rigor. You say there's no one answer to questions about which possible worlds are nearest? Lewis would agree. He'd point out, however, that once you're settled on the criterion of closeness that fits your purposes, you can apply his apparatus.
A closing thought: suppose the reply to my comments is that I still haven't addressed the issue about applying rigor to the non-rigorous. (It's not clear to me that your original worry amounts to this, but no matter.) Even to apply Lewis' apparatus contextually goes beyond anything we can do with absolute rigor, or so it could be argued. But now the criticism proves too much. We're almost never in a position to apply physics (or any other science, for that matter) with the sort of rigor that criticism has in mind.
Theories in philosophy are often like theories in science, or so I'd suggest: they're more or less useful intellectual tools. My own take on what Lewis and Stalnaker have bequeathed us is that this intellectual tool has more than proved its usefulness. That's not to say it's beyond criticism or will never be replaced. But it's a considerable accomplishment.
One point worth noting here is that words like "fake" are, so far as I can see, always intensional. meaning that whether something is a fake F depends upon what property F is, and not just which things are F. They are also "attributive", meaning that an Adj-Noun isn't just an Adj that is Noun, but (roughly) something that is Adj for a Noun. E.g., a tall basketball player is someone who is tall for a basketball player, not just someone who is tall and a basketball player. Attributives are hard enough; intensionality is hard enough; both by themselves. Put them together, and it's a nightmare.