Eggs from Gifts

Not that I'm aware of. Assuming I've cleared my egg storage I seem to get an egg roughly one out of three gifts.

I'm hoping they add a bit more variety to the gifted eggs soon.

Not that I am aware of.
One of the first days I got on egg on almost every gift up to the gift limit 20. A total of 18 eggs.

Be sure to check your storage!
Sometimes, especially if you use a Go Plus, a normal egg have sneaked pass you and fill the last free slot.

I have ran with 9 incubators since the start. Usually get at least nine gift eggs a day, but streaks with 4-5 gifts without one frequently happen.

There's no limit I don't think but the egg drop rate seems to have gone down. I walked all my eggs yesterday so I could refill with all alolans today and in the 20 gifts I opened, I got 6 eggs. Had to change my time-zone so I could open around 5 more gifts to get the last 3 eggs.

You have to make sure you have an open egg slot and that your Pokemon storage is not full. Eggs count as Pokemon, so if your storage is full, no egg will drop.