Serious questions about time travel in "Kate & Leopold"

So here we go again.

Is the movie depicting a fixed timeline with a causal loop within (like 12 Monkeys), or is it an alterable/malleable timeline like Back to the Future?

The errors/debatable points are:

(BIG SPOILERS)

Premise: assuming it's a FIXED TIMELINE/CAUSAL LOOP.

1- How did Stuart (Liev Schreiber) manage to take a photograph of Kate (Meg Ryan) at the party in 1876, if she arrived at the very end, when Stuart wasn't there anymore to take any photos?

2- In the final sequence of the movie, Leopold (Hugh Jackman) still stays at the party, like he never left it. But he must discover/chase Stuart in order to experience his adventures in 2001 and meet Kate for the very first time. Remember, it is supposed to be a loop.
So what?

Is "the" Leopold at the very end "the" Leopold #2 that returned back to 1876, (hours or minutes) AFTER the "first" Leopold #1 had left to chase Stuart? This way, the loop should be preserved.

Tell me. Rewatch the movie. Thanks.