Sounds like you are in a bit of a sticky situation, did you let your employer know when you applied for the job what your areas of expertise were around? Did they just put you as a font end developer knowing this?
The reason why I ask is that there may have been some confusion around what role you wanted to do and the role of the font end developer.
My advice would be for you to sit down and have a candid discussion with your boss. Mention the fact that you have been trying to teach yourself coding, as your background is design, and that you have been struggling.
Are you the only front end developer?
See if you can make a case for why you could do the UX and UI work instead, this can be great practice for later jobs. When having discussions it is good to mention the difficulties that you are coming up against, but it is even better to have possible solutions. So maybe come up with a game plan of all the things you can do in the UX area, and how they would be beneficial to the startup. As really, they need to be integrated from the start, including user research. After you have come up with a plan, go have the chat, and you may find that you get the go ahead.
But this is coming from a standpoint of not knowing much of your team, the roles, etc. So for instance, if you are the only front-end-developer, then your boss may not be as keen for you to move away from the coding (although personally UX work should be done before writing code).
I've found that people find it a lot easier to say yes they like things (or don't), yes that's a good idea (or not). So if you show initiative, suggest a plan, try and prove to them that having a workflow and process will benefit the team, and therefore whatever you are trying to create, they are much more likely to get on board.
I think that the requirement to be able to code is becoming much stronger, but again it depends on your team size. I'm guessing that with startups, there may be a lot more mixing of disciplines because of the smaller teams. In the job the I currently work in, I am a User Experience Designer, and I am not required to code. I'm learning web for interest and personal use :). But I also work in a bigger company where they have front-end and back-end developers.
Feel free to ask any more questions!