Would I need to varnish the floor again after repairing the gaps?

Yes. But before you get involved in this project, back off and wait a few
months for more serious house faults to crop up. The sellers obviously did a
hack job on the floors to make the house marketable, so you need time to ferret
out other shortcuts they have taken.
For now, put down low cost area rugs to keep the flaws out of sight while you
become accustomed to your new quarters. If you need something to do, research
ways of paint removal for the other problematic floors if thay seem good enough
to shape up with a standard refinish regimen.
It is possible that removing the old flooring and installing new is the way to
go. Get per foot estimates from local suppliers for hardwoods or laminates
depending on your preferences, then go for best value. HTH

Joe