Learn Russian: Он безуспешно пытался, но всё-таки не сдался. - He tried unsuccessfully, but nevertheless did not give up.

While но simply connects the two clauses with 'but', adding всё-таки emphasizes that the second action happened contrary to expectations. Since his attempts were unsuccessful, you would expect him to quit. Всё-таки highlights the 'nevertheless' or 'despite that' nature of his refusal to give up.

Could we place всё-таки somewhere else in the second clause?

Yes, particles like всё-таки are quite flexible. You could say но не сдался всё-таки or add the pronoun again: но он всё-таки не сдался. Placing it immediately after но makes the contrast punchy and immediate.

Why is the first verb imperfective (пытался) but the second verb perfective (сдался)?

Пытался (tried) is imperfective because the effort was an ongoing or repeated process. Сдался (gave up) is perfective because giving up is a single, definitive action. By negating the perfective verb (не сдался), we are saying that this final, conclusive event did not occur.

Why does сдался have the reflexive -ся ending here?

Without the reflexive ending, the verb сдавать / сдать means to hand something in (like an exam or an apartment). Adding -ся literally means 'to hand oneself over', which translates to surrendering or giving up.