Do you feel that philosophy suffers from a lack of respect from the public and
It's tough to get a handle on whether philosophy is respected by the public. First and foremost, my guess is that most people living today don't respect or disrespect philosophy. They simply have no attitude toward it whatsoever. In large measure, that's because most people are exposed to philosophy through university education, and only a small minority of people receive such educations.
It's also important to distinguish philosophy the practice from philosophy the academic discipline. One could respect the former and not the latter, and vice versa.
Finally, attitudes toward philosophy seem to vary from community to community (it seems more respected, in my observation, in Europe than in the U.S.).
So I'm reluctant to make any sweeping statements in response to your question. But here are some relevant (and contestable!) observations:
1. The world is thirsty for philosophy. I think that those who are exposed to philosophy see that it engages questions that impact themselves and their communities in a deep way. It's not a discipline that asks esoteric questions.
2. That said, philosophy as an academic discipline is very off putting to many. To some extent, this is because academic study of anything is off putting to many: It requires people to do things they are often disinclined to do, like reading carefully, grasping opposing positions, etc. But philosophy also requires a rare form of patience. Philosophers are suspicious of snap judgments, but many who encounter philosophy assume that philosophy is like religion, a dispenser of truths. Philosophy nurtures hard questions, not easy answers.
3. Because of point 2 above, philosophy is often 'dissed', even by people who should know better. (There have been several instances recently of prominent scientists declaring philosophy dead or irrelevant based on, well, shoddy philosophical thinking!)
Long story short-ish: I can't say whether philosophy is respected or not, but it would certainly behoove the discipline to be better known and better respected. And there's blame to go around. Many people don't understand the aims of philosophy, but the philosophy community has often not been particularly good at helping people see those aims and, as a result, has not earned the wider respect of the public.
Fortunately, things are changing on that front. Increasingly, academic philosophers are encouraged, even expected to engage with those outside their academic circles. 'Public impact' is now taken more seriously than it was in the past. As a result, 'public philosophy' has spread rapidly. Here's a list of recent op-ed winners to illustrate my point: http://www.apaonline.org/news/248593/Public-philosophy-op-ed-contest-win...