If my husband inherites the house that we are living in, do I have any legal rights to the property or the money that we paid to lease to lease it?

As some sort of an inheritance? No.

If you were to seek part of the house in a divorce? Perhaps.

Are you getting divorced? I'm not sure why it would matter if you are not getting divorced. However, as said above it is not your inheritance so in that means, no you have no rights it can be placed in his name only. But being his spouce if there are no plans for divorce you won't be forced to leave either.