As I'm running a straight edge perpendicular along my joists, I'm noticing there are gaps of about 1/8" on some of them. Is this an acceptable tolerance?
Joist to joist they are 1/8" out of level? That is pretty typical. You always want to install joists with the "crown" up and 2x4's in the wall with the crown all facing one way. When dealing with wood differences are to be expected. Steel studs are much truer than wood ones, though they are not structural they only support the drywall not the structure above.
If you use Trex or other composite decking you'll really notice that the deck joists aren't even. It doesn't show up as much with wooden decking but the joists are similarly uneven.
Any advice given here is general in nature and is not necessarily valid for your given area. If in doubt check with your local codes enforcement department for what is required when doing electrical, plumbing or structural work on your house. Permits may or may not be required in your area and home owners may not be able to DIY some tasks. I have no way of knowing if you have the skills needed to complete the tasks you are asking about, when in doubt seek professional assistance.
My advice may be worth exactly what you pay me for it. :-) For the record I did not stay in a Holiday Inn Express last night.