I've declared my attack, and my opponent declares their blocker. I then use an instant that boosts and gives first strike to my attacking creature, then in the middle of combat they tap the creature to activate its ability. Can they do this?
Sure. As long as the creature is able to block at the beginning of the declare blockers step (typically this means untapped) it can be declared as a blocker. What happens after that doesn't affect it's status as a blocker. There are also no restrictions on whether it can tap to use an ability, pay a cost, etc.
A blocking creature may be tapped to pay a cost or as the result of an effect.
This has no effect on its status as a blocker. It will still deal and receive damage as normal.
It can't be tapped for no reason.
The only way to stop a creature from being a blocker before the end of combat is to remove the creature from combat.
506.4. A permanent is removed from combat if it leaves the battlefield, if its controller changes, if it phases out, if an effect specifically removes it from combat, if it’s a planeswalker that’s being attacked and stops being a planeswalker, or if it’s an attacking or blocking creature that regenerates (see rule 701.13) or stops being a creature. A creature that’s removed from combat stops being an attacking, blocking, blocked, and/or unblocked creature. A planeswalker that’s removed from combat stops being attacked.