Some parts of the movie I would want to change

Fight scene in beginning:
I guess they wanted to show us this Lara is not yet the Tomb Raider, being defeated by a similarly small no-name girl. I'm ok with this in an origin story, however in this case they should not show her grappling with powerful mercenaries mere weeks after the scene.
Instead they should have had her avoid this kind of fighting, maybe making a comment in line with "Not getting myself in another headlock", then proceeding to use her wits and defeat her adversaries from a distance. Resorting to dirty fighting when cornered.

Not signing the documents initially:
Lara is supposed to be intelligent. She knows signing these papers makes no real difference on her dad being truly alive or not. Her search and rescue operation would have a lot more options if she signed the papers and acquired her inheritance. Instead of pawning jewelry to rent a shabby boat for going into treacherous waters, she could just get a helicopter and some armed personnel to go with her.
She could still visit the captain for directions and possibly enlist him, but when he says something about the sea being too treacherous near this island Lara could reply "Who said anything about going by sea?" as the helicopter makes its approach.
We could still have basically the same storyline afterwards, if her security detail was overpowered or (mostly) infiltrated by Trinity.
Her dad might have used the ship as a way to reach the island without alerting Trinity, but for Lara this is not a reason since she has no idea beforehand that they have a presence there.

Lara should also get the list of corporations before the island, realizing mid story that Ana is a traitor and taking her down in the final scenes instead of leaving this part open.