Learn Hardcore Italian: Leggo qualsiasi libro che trovo in giardino.

qualsiasi means “any, whichever,” conveying indifference to which specific book.
ogni means “every, each,” emphasizing all books individually.
So:

leggo qualsiasi libro = “I read whichever book I find.”
leggo ogni libro = “I read every single book I find.”

Could I use qualunque instead of qualsiasi?

Yes. qualunque is a direct synonym of qualsiasi in most contexts.
For example: leggo qualunque libro che trovo in giardino has the same meaning and nuance.

Why is the verb trovo in the indicative mood and not the subjunctive?

The subjunctive in Italian is triggered by doubt, desire, or hypothetical situations. Here, trovo describes an actual, factual action (“that I really find”), so the indicative is correct. If the clause were purely hypothetical or non-real (“soffo che trovi” doesn’t make sense), you would consider the subjunctive.

Why do we say in giardino instead of nel giardino?

in giardino (no article) is the usual way to express being “in the garden” in a general or habitual sense.
nel giardino (in + article) would be used to specify a particular garden you’ve already identified (“in the garden we talked about”).