Marc, there is no magic in lawn care. Simple routine maintenance is the key. Mulch mow at the higher cut range for your grass, fertilize, and water to 1" per week. Your climate may be a factor too. I am not at all familiar with Alaska growing conditions so I don't know how much of the lower 48 techniques work up there. I would not worry about lime until you get a soil test. Is there a local soil conservation office that offers soil testing? UAF has a cooperative extension site and normally that is the best source for local growing conditions and techniques.

Clover is not necessarily bad. Your choice if you want to keep it.

For the 6x10 area, rake as much of the gravel out as you can. Core aerate and overseed should get you going with some grass.

+1 Lawncare isn't rocket science, but it is a bit of work and you need to do it consistently. Mowing is #1. You really have to do it at least once a week in the growing season. You won't want to let it get tall and then chop it way back. You definitely need to mow before winter sets in. Snow and ice mat down the grass if it is too long and you definitely want to remove or chop up any leaves before they smother the lawn.

For the gravel area, I suggest you dig it all out. Remove all the stone and dig down at least a foot. Mix some compost in with the native soil, level, and reseed. You will want to leave it slightly mounded because the soil will settle a bit after digging it up.