Should I ask him how he feels?

I am not sure if I have feelings for someone because we went out on one date and I am questioning if it was as memorable for him as it was for me. I felt as if I had met someone who understood me like few people have before, which is why I was surprised when there was no message or question about a second date. I was very sure that he would get in touch soon and ask me out, but it has been two weeks and there has been nothing. I am tempted to call and ask him if he didn’t have a good time, but this may frighten him off if he thinks I am too clingy. What should I do? 
You should wait for a while and avoid jumping to conclusions because you know nothing about what this person is going through. It’s possible that he may not share your sentiments about the date, but assuming the worst is unfair to him as well as you. If nothing happens for a few more days, you can choose to continue waiting or message and ask him out because that is a clear indicator of where his interests lie. It isn’t a question of coming across as clingy because you are allowed to reach out to an adult and ask them out.

How will I ever get over the idea that my partner is not being entirely honest with me? He hasn’t done anything specific, but it is his nature which is secretive, and which makes me wonder if there is something about his life that I know nothing about.
If there is no reason to be suspicious, you are harming this relationship by being unable to accept your partner. If you have questions about something, you should ask them because not doing that is bad too. Building a strong relationship built on mutual trust is critical but takes time. Speak to him about what’s on your mind and ask for clarity whenever you need it.