#1. See what's going on - you do NOT want birds, squirrels, raccoons, etc. in your chimney. There are caps that can be installed that will prevent this in future. This does not mean you need the chimney cleaned, just checked. This is an expense you should cover - the tenants have no control over critters getting in.
#2. What does "bugs" mean? (First, you should have some signed acknowledgement from your walk-through that the premises were bug-free at move-in. If not, you may want to add that to your paperwork for the future.) Do they want everything sprayed "just in case" something decides to come inside? Or do they have a specific problem: roaches, fleas, silverfish? Your lease should have a clause that specifies who pays for extermination; i.e., if there weren't roaches (for example) at move-in, the tenant pays.
In a SFR, (at least in these parts) the tenant assumes many of the maintenance duties & expenses that in a multi-family would be covered by the owner. If you do allow them to hire an exterminator, I would insist that you have final approval of who does it & how. Make sure they're licensed, etc. & try to be there when they do the application.
If you decide they just need to set off a couple of bug bombs, remind them that any damage to carpets, drapes, walls, etc. will be coming out of their deposit.
And of course, make sure all the window screens are in good shape & they're not leaving the doors open! Check your exterior caulking, soffits & siding & make sure everything is bug-tight. CYA - or CYHouse, I guess. Good luck!