I have cuttings rooting right now, and there are about 4 that I would like to put into pots so they can be buried in ground during the warm months. For the first year, what would be a minimum size in gallons. I have lots of 1 gallon pots, but is that big enough?

Agreed.  If you are burying them, 5 gallon pot would be minimum.  That should last for your first growing season in zone 5, but expect the pot to be root bound, even with holes cut out of the side of the pots.  Figs grow like weeds!  :)

Growth rate is often more determined by the variety. I have some in #3 pots for 3 years. Others I have shifted up to #15 in 7 months. It really depends on the variety of tree. To be pretty safe I'd say 10 gallon. I would not plant new trees in the ground where it gets that cold for at least 2 years. Store the pot somewhere for 2 winters and then plant in the ground.

Even in a 10 gallon pot you will be chopping roots away when you lift out the pot for winter storage.

Scraping a tree and seeing green in the winter does not necessarily mean the tree is viable. My Christmas tree is still green, even though it's dead. They just take a while to show it.

When sinking pots in the ground would I be correct in assuming you need to leave out any sort of drainage layer (gravel etc) at the bottom of the pot so the roots can work their way through the drainage holes? I'm using 6 gal buckets. I think my plan will be to drill all my drainage holes at the bottom sides of the bucket rather than underneath so I can cut them free more easily at the end of the season.

Also wondering how others are changing the potting mix as they go from the light rooting stage to the more long-term pot? Are you using heavier mixes that hold more water or nutrients? Last year got a couple of 2 yr old trees. I potted them up to 5 gallon pots (buckets). I used 3 parts potting mix and added in 1 part sheep manure and 1 part sand. The foliage seemed really healthy but no fruit to speak of. I put that down to it being a young tree. Maybe there's something I could have done better. I'm wondering about mix for the next stages.

Putting the holes at the bottom sides is great advice! I didn't even think about how the heck I would get under to cut the roots. That saved me a whole lot of work!

started at 1 gallon then must up pot to 3. after 3 gallon id prefer 7 gallon. dont think you can get away with 1 gallon all summer unless problems happened. after experimenting big holes in side of pots is best. holes on the bottom are harder to dig up for winter. a shovel makes easy work of roots growing out the sides. i rather have my roots grow out of the pot over using a self watering pot. i use wood chips around pots which keeps plenty of moister. plus this makes pulling the pots easy.