Consistent where doing that only works because a future time traveler took my wallet so I just end up with one (Self consistent aka Harry Potter time travel).
And one way which basically means if I go back there are now two of me and I have to go forward the slow way. Which is separated from the first example by erasing the future when you travel back.
Paradoxes are only an issue in the second example.
As to other interactions their are a great many possibles but they don't mean much. For example restricting things so only light can move though time does not change the underlying options in terms of information and paradoxes.
PS: Mathematical rigor is again a human idea, our universe could be absolutely random and we would have no way of knowing. At best we can say that seems unlikely.
It's how the time machine worked in Timeline.
If I remember correctly, the company sending people back through time (really to a parallel universe) didn't even know how/why the receiving machines were appearing.
It also happened on Sliders, they visited worlds that were ahead or behind Earth Prime.