Is there any scientific proof that one's mental state influences their educational success?

Yeah, if you can ignore the obvious flaws with the terms "proof" and "mental state."

https://pediatrics.aappublications.org/content/124/4/e596.abstract

[quote]Teens with depression and their parents reported more impairment in all areas, compared with teens without depression at baseline, and reported more coexisting emotional and behavioral problems. Both depression and coexisting problems were related to impairment. There was a lasting impact of depressive symptoms on most measures of peer, family, and school functioning 6 months later, but controlling for coexisting baseline emotional and behavioral problems attenuated this relationship for some measures.[/quote]

It seems obvious to me, that if you are mentally healthy and balanced, you would be able to focus and retain information better, be more motivated to learn, etc.

There are some convincing meassures about.
And some espectacular but isolated exceptions.
Seems probable that what affects enough our nervous system will also affect a specific behaviour of it, like learning.

I couldn't finish college my first attempt because of severe untreated depression and anxiety.

"When you think of clinical depression, you probably think of feeling sad and down for long periods of time; losing your energy and your interest in things you used to enjoy; sleeping too much or too little, or eating too much or too little. But besides these, depression can actually change your ability to think. It can impair your attention and memory, as well as your information processing and decision-making skills. It can also lower your cognitive flexibility (the ability to adapt your goals and strategies to changing situations) and executive functioning (the ability to take all the steps to get something done)."

An excerpt from a paper done by James Cartreine, PhD for Havard Health Blog. As for other proof and statistics? I'm sure you can find plenty of info.

I guess being depressed or mentally ill would make it difficult to study or focus or have the inspiration to even want to do that

Yes! Look at three-quarters of SW's members! All expelled from kindergarten!