Squick: So, Stuart slept with someone who, as it turns out, is his grandmother? Boy, that's gonna be awkward when he's, uh, born...
There was originally a scene along the lines of Stuart and Kate realizing the connotations and being thoroughly grossed out, but TPTB decided to leave it out of the film and have that particular point left as Fridge Logic. Probably for the best.
One might infer that since Stuart doesn't know Kate turns out to be his grandmother, that she died either before he was born or when he was very young. I hate temporal mechanics.
She wasn't his grandmother - she was his great x3 grandmother. After all, she was in her 30s in 1876, and it was expressly stated at one point that Leopold is his great x3 grandfather. As for why he didn't know she was his grandmother, the text he refers to at the beginning of the film doesn't refer to Kate by name - only referring to her as "the love of [Leopold's] life". After he found out that she was supposed to go back in time, then he had more important things on his mind, i.e. getting Kate to the portal in time. It probably wouldn't be until he gets back to his flat, and has a think, that he'll realise - this is probably why the scene with him realising is cut.
Ok, so the elevators disappear when Leopold does. But why are the elevator shafts and doors still there? Would the architect put them there for aethetic reasons? Step-by-step changes are as stupid here as they were in the Film of the BookA Sound of Thunder. Gotta think in terms of cause and effect. Then again, this is a romantic film, not a sci-fi movie.
On another point why didnt some other guy invent the elevator its not a particularly hard idea to come up with you just need to make it safe.
Because Leopold invented the elevators, not some other guy. That's what happens if you take an inventor from his time (before he invented anything) and moved him to this century. Because, in the time-stream, he didn't invent the thing, they start disappearing.
Which would be fine and dandy if Leopold, Duke of Albany had actually invented the elevator. In reality, the earliest iteration of the modern elevator was invented by Elisha Otis.
Also, the vanishing elevators still don't work, since removing the supposed inventor of the elevator only ensures that he doesn't invent it. It doesn't remove the chance that someone else in the vast time between then and the present would have invented a similar device to fill the need. There are many real life stories of concurrent invention by different people - the first to register (oversimplification) getting the credit.