Probably to the second, maybe to the first.
Clearly someone can be Secret-Keeper for a dwelling that they themselves spend time in. Bill , for example, is the Secret-Keeper for the location of Shell Cottage, where he resides.
“How are they protected?” asked Harry. “Fidelius Charm. Dad’s
Secret-Keeper. And we’ve done it on this cottage too; I’m Secret-Keeper
here. None of us can go to work, but that’s hardly the most important
thing now. Once Ollivander and Griphook are well enough, we’ll move
them to Muriel’s too. There isn’t much room here, but she’s got
plenty. Griphook’s legs are on the mend. Fleur’s given him
Skele-Gro—we could probably move them in an hour or—"
There can certainly be multiple Secret-Keepers for a given secret.
Mr. Weasley had explained that after the death of Dumbledore, their
Secret-Keeper, each of the people to whom Dumbledore had confided
Grimmauld Place’s location had become a Secret People in turn.
“And as there are around twenty of us, that greatly dilutes the power
of the Fidelius Charm. Twenty times as many opportunities for the
Death Eaters to get the secret out of somebody. We can’t expect it to
hold much longer.”
It would seem as if there is nothing standing in the way of Lily and James being Secret-Keepers for the same location. At the very least, if say, Dumbledore were the Secret-Keeper for their location, and they were the only other ones who knew of it, then if Dumbledore died they would presumably become co-Secret-Keepers.
Why did they then choose someone else to be their Secret-Keeper?
Perhaps they wanted to go out occasionally, and neither of them wanted to be able to reveal the location of the other (assuming that torture or threats might work). Certainly they chose Pettigrew because he would be less conspicuous.
But yes; it seems likely that for two people who never want to leave their house, they could both be Secret-Keepers for the same location.
With regard to the first instance, nothing seems to stop the sort of recursive Fidelius Charm mentioned, but on the other hand I know of no statements from Rowling that say it is possible.