Learn Hardcore Russian: Даже когда я устаю и чувствую себя уставшим, я всё-таки стараюсь выучить хотя бы одно слово. - Even when I get tired and feel exhausted, I still try to learn at least one word.

я устаю = I get tired / I become tired (the process or state of getting tired).
я чувствую себя уставшим = I feel tired (my subjective perception of my state).

Together, they sound natural and slightly emphatic, like in English:
“Even when I’m tired and feel exhausted…”

You could say only one of them and still be grammatical:

Даже когда я устаю, я всё-таки стараюсь…
Даже когда я чувствую себя уставшим, я всё-таки стараюсь…

Using both just reinforces the idea of tiredness and sounds very natural in Russian.

Why is it чувствую себя уставшим and not чувствую уставшим or чувствую уставший?

The structure here is:

чувствовать себя
- (instrumental case) = to feel (oneself) + as / in a state of ...

себя is the object of чувствую (I feel myself), and уставшим is in the instrumental case, describing the state:

чувствую себя уставшим = I feel (myself) tired.

Compare:

чувствовать себя больным – to feel ill
чувствовать себя счастливым – to feel happy

So:

чувствую уставшим – wrong (missing себя).
чувствую себя уставший – wrong case; should be instrumental уставшим, not nominative уставший.

Why is уставшим in the instrumental case?

After чувствовать себя (and many verbs that mean to be/feel in some state), Russian uses the instrumental case to describe that state:

чувствовать себя уставшим – to feel tired
быть студентом – to be a student
оказаться неправым – to turn out to be wrong

So the pattern is:

чувствовать себя
- кем? / каким? (instrumental)

That is why уставший becomes уставшим (masculine singular instrumental).

Why is it я устаю and not я устал in this sentence?

Both are correct but have different nuances.

я устаю – present tense of уставать (imperfective):
- describes something habitual or regular: I (tend to) get tired / I get tired (whenever this situation happens).
я устал – past tense/perfective result:
- a concrete, finished result: I have become tired / I am (already) tired.

In Даже когда я устаю…, the speaker talks about what usually happens in such situations, so the present habitual form устаю fits well.

If you said:

Даже когда я устал, я всё-таки стараюсь…

it would sound more like: Even when I have already gotten tired (right now / in a particular situation), I still try… — more specific, less general.

Why is the verb устаю in the present tense after когда, instead of something like a future form?

In Russian, the present tense with когда is often used for:

general truths, habits, and repeated situations.

So:

Даже когда я устаю, я стараюсь…
≈ Even when I am / get tired, I (still) try…

Russian doesn't need a special "present continuous" or future here; the simple present устаю already covers the idea of whenever I (end up) tired in such cases.

Using a future form когда я устану would refer to a more specific future moment:

Когда я устану, я пойду спать. – When I get tired (later), I’ll go to sleep.

Why do we need себя in чувствую себя уставшим? Can we just say чувствую уставшим or я уставший?

чувствовать себя means to feel (oneself) in some state and is a fixed pattern. You almost always need себя here.
- чувствую себя хорошо / плохо / уставшим / больным

Without себя, чувствовать usually means to feel (something) physically or emotionally:

чувствовать боль – to feel pain
чувствовать радость – to feel joy

So:

чувствую уставшим – ungrammatical in standard Russian.

But you can say:

Я уставший. – I’m tired. (adjectival predicate)
Я устал. – I’m tired / I got tired. (verb, past tense)

They are simpler, but they don’t use the чувствовать себя construction.

What does всё-таки add to the sentence? Could we leave it out?

всё-таки roughly means still, nevertheless, even so, after all. It emphasizes that the action happens despite something.

я стараюсь выучить… – I try to learn…
я всё-таки стараюсь выучить… – I still / nevertheless try to learn…

So:

Даже когда я устаю…, я стараюсь выучить… – neutral.
Даже когда я устаю…, я всё-таки стараюсь выучить… – highlights that it’s difficult but the speaker makes an effort anyway.

You can leave всё-таки out; the sentence will still be correct, just a bit less expressive.

Where can всё-таки go in the sentence? Is я всё-таки стараюсь the only correct option?

Russian word order is flexible, and всё-таки can move, though not all positions sound equally natural. Common options:

…я всё-таки стараюсь выучить… – very natural.
…я стараюсь всё-таки выучить… – also natural; a bit more focus on managing to learn it after all.

Less natural or marked:

…я стараюсь выучить всё-таки хотя бы одно слово. – possible, but всё-таки usually stays near the verb or clause it modifies.

You normally would not place it at the very beginning here:

Всё-таки, даже когда я устаю… – possible in some contexts, but it changes the rhythm and emphasis and is less neutral.

Why is выучить used here and not учить or учиться?

These verbs are related but different:

учить (что?) – to study / to learn (in progress), or to teach (кого?).
- учить слова – to study words.
учиться (где? как?) – to study (somewhere), to be a student.
- учиться в школе – to study at school.
выучить (что?) – to learn something completely, to master/memorize it. Perfective aspect.

In the sentence:

стараюсь выучить хотя бы одно слово

the idea is to manage to fully learn at least one word (so that I know it). That’s why the perfective выучить is used: it focuses on the result, not on the process.

If you said:

стараюсь учить хотя бы одно слово

that would sound more like I try to be in the process of studying at least one word, which is a bit odd; with such a small, concrete object (one word), выучить is more natural.

What exactly does хотя бы mean here, and how is it different from по крайней мере or как минимум?

хотя бы means at least in the sense of even if it’s only this minimum, that’s already something. It often has a slightly modest or hopeful tone:

выучить хотя бы одно слово – to learn at least one word (I’d be satisfied even with one).

Rough differences:

хотя бы – at least (as a minimum I’d be glad to reach)
по крайней мере – at least (as a consolation or correction)
как минимум – at least in a more neutral, quantitative, often more formal way.

Examples:

Сделай хотя бы это. – Do at least this (I know it’s not much, but it would already help).
По крайней мере, он извинился. – At least he apologized (even if other things were bad).
Нужно выучить как минимум десять слов. – You need to learn at least ten words (requirement / fact).

In the original sentence, хотя бы is natural because the speaker is setting a minimal, modest goal.