Epistemic status: Hardcore projecting myself onto a stranger.
I was in college pretty recently, and I think I recognize in this question a lot of the same unhealthy attitudes that were so toxic for me in college and for the year(ish) after graduation. Like this:
I feel like I'm unable to let go of the fun-loving part of me which needs stupid entertainment. I simply cannot stay interested enough in learning and knowledge to be doing it 24/7, but I feel like this is requisite in order to be a successful thinker.
This is just not how life works. The vast majority of people, including the really successful ones, like "stupid entertainment" of one form or another. Habryka watches a lot of YouTube. Luke Muehlhauser is obsessed with corgis. Elon Musk.... smokes weed on live TV. It's not intrinsically bad to enjoy things that aren't work.
You are framing this as "I'm unable to let go of the fun-loving part of me." I think that's dangerous. Interesting and successful people still enjoy hanging out with their friends and doing things that aren't work. Staying interested in one single field 24/7 is definitely not a requisite for being a successful thinker, and in fact is probably counterproductive (see David Epstein's great book Range on this subject). Keeping yourself happy and not burned out is really important, and following your curiosity to a variety of other fields can often give you valuable perspective on your core work.
How do you guys stay interested in something (an idea or even an entire field) persistently enough to always be motivated to work on it?
(The following paragraph is probably fairly specific to the existential risk community (as compared to e.g. academia), but you did ask on LW, so, y'know. That's what you get.)
For most of the people I know who are doing really intense work, they don't stay motivated solely out of 'interest.' If Buck Shlegeris were just following his interest, he'd likely spend more time on physics and music than he does, but instead he devotes a lot of his time to MIRI because he believes in the importance of working to reduce existential risks. That's not to say he doesn't enjoy his MIRI work, just that it's not all about "staying interested." Sometimes we do things because we endorse doing them, rather than because we just want to do them. I've heard of some rationalists who claim to have integrated all of the subcategories of their personality (to use your term), but these people are by far the exception rather than the rule.
Is it unrealistic to hope to always be motivated by your curiosity?
Yes and no. There might be times when you're just devouring everything you can on a topic – I remember in high school I used to spend Sundays at my friend's house with all the other girls in my calculus class, doing extra credit work for fun, and then I would go to math team competitions after school and talk with my friends about proofs at lunch. I think there are academics who are also like this – in particular, some professors seem to just want to talk about their field all the time, and they seem to really enjoy it. Maybe it's possible to intentionally cultivate that level of sustained enthusiasm, but if so I don't know how to do it, and I wouldn't count on it as your only motivator. Curiosity can drive your choice of field and keep you excited about it on medium timescales, but not minute to minute.
I like my job quite a lot, but there are plenty of days when I don't feel intrinsically motivated to do it. Days when what I really want is to do housework or practice some song on the guitar or go for a long walk in the forest. But I do my work anyway, because I've committed to do it – because there would be consequences if I just didn't show up to work, because my coworkers (who I really like) would have to shoulder the burden I left, because my financial security is tied to it. Curiosity is a lovely motivator if you have it, but external commitments are much more reliable.
Will I burn myself out if I devote my free-time to extracurricular reading?
Not if you still allow time for other things that provide you with value! (See the recommendation of goal factoring below.) And especially not if you read because you're following your interest, rather than because you think you 'should' (see also p.167 here). I read like it's a religion and it often gives me energy rather than draining it. I'm a 'technical writer for software in the streets, rationalist in the sheets' with a degree in physics, but I read about whatever I want – currently that's mostly urban design, nutrition, and evolution. I love reading. But if I'm not into a book, I'll drop it. I think you should generally not perform mental violence in order to get yourself to do things... although being in school probably makes that hard.
Recommendations: A fair amount has been written on LW about the value of rest; see the Slack and the Sabbath sequence for a good start. I also recommend looking into CFAR's technique of goal factoring, where you try to get at the reasons why you're really doing something. (See also the Hammertime post and the CFAR handbook). Not to write the bottom line for you, but I expect you'll find that things like hanging out with your friends are providing you with value that you couldn't get by spending all your time studying.
Scott Alexander's wanting vs. liking vs. approving framework also seems relevant here (though, spoiler alert, it's kind of a confusing mess if you actually try to pin down what he means by each word.)
Also extremely relevant: Eliezer's On Doing the Impossible.
College is a unique environment where you newly have a lot of control over how you spend any given moment of your time, and yet you have more work to do than can actually fit into that time. (This was often literally true for me, YMMV.) You pressure yourself because your grades aren't as perfect as they were in high school, and you feel like you could be doing more if you were just somehow better. But I think that comes from a flawed orientation towards your goals. Instead of torturing yourself over those lost hours of studying, I recommend half-assing it with everything you've got. There are some classes where trying to get value out of them is like drawing blood from a stone, so maybe don't bother – if you're interested in the material, just read on your own. Unless you're shooting for grad school (which maybe you are, and I don't know much about that world), as long as you keep your GPA above 3.0 and get that piece of paper two years down the line, no one's going to care about anything else. Don't feel like it's wrong to live your life in a way that makes you happy. Don't beat yourself up too much. And good luck <3