As someone with a similar background as yours (I was an international CS student who first learned programming in the freshman year at a English-speaking college), I can only say that you are not alone in this struggle. If I can give an advice to my younger self, I would ask myself not to beat myself too much for not being able to be as smart as the people around me. In short, don't compare yourself to others, because it gives more harm than good. Instead, appreciate the good thing you have done and keep doing your best at whatever things in front of you.

> I tend to make a lot of syntax errors. I also tend to rely on documentation a lot.

Believe it or not, many of us who work in the software industry rely heavily on IDE to fix syntax errors. My suggestion here is try to pick your favorite language and keep practicing (and make mistakes) until you're comfortable using it. I hate to say that there is no shortcut in getting better. Small mistakes are tolerable, even when you're doing whiteboard interview later on. More importantly, I would say that understanding concepts in programming languages is a LOT more important than remembering syntax (i.e. memory management in C, heavy emphasis of OO in java, functional programming in scheme, and so on).

As for relying on the documentation, it does not come across as a bad thing to me. For example, before I call a function from a 3rd party library, every so often I check the documentation for it. From there I can learn the contract of the code or some gotchas that I need to be aware of.