Soon I will be lifting and dividing my tulips bulbs. Should the offset bulbs be put back into the ground now or in the Fall?
Either would work fine.
It is probably be a lot less work just to plant them now and not worry about them rotting or drying out in a bag until Fall.
Some of these bulbs are very tiny without any root at all. If I plant them now will they grow some roots over the Summer or do they just stay dormant?
If you plant them like a row of peas and put a marker at either end of the row.
One tip I read of says to plant them onto grit (not sand) that is a different colour from your soil. When you see that as you're lifting the bulbs you know to proceed with care to avoid bulb kebabs.
They often start into roots toward the end of summer - late July/early August, though it may be later if you have a long dry spell around then.
Give them two to three years to make it to flowering size - and feed them while they're in leaf.