Я открою запись вебинара позже и переслушаю её в наушниках.

I will open the webinar recording later and listen to it again on headphones.

Breakdown of Я открою запись вебинара позже и переслушаю её в наушниках.

• я: I
• в: in
• открыть: to open
• и: and

Questions & Answers about Я открою запись вебинара позже и переслушаю её в наушниках.

Why is Я included here? Can Russian drop it?

Yes, Russian often drops the subject pronoun because the verb ending already shows the person/number.
Открою … и переслушаю … already means I will open … and (I will) listen again …, so (Я) открою… is also natural.
Including Я adds a bit of emphasis/contrast (e.g., I will do it, not someone else).

What tense is открою and переслушаю? Why is there no separate word for will?

Both are simple future forms. Russian doesn’t use a separate auxiliary like English will for most verbs.
Because открыть and переслушать are perfective verbs, their “present-looking” conjugation actually expresses future:

• открою = I will open
• переслушаю = I will listen again / re-listen

Why are these verbs perfective? What does that imply?