As an IC basically my entire life, I have joined large legacy projects many times, and my typical attack strategy always revolves around a large cup of coffee and my trusty debugger. Using 2 monitors, starting at main() or index.php (or whatever of course), I will execute as much of the entire codebase as possible line-by-line.
At first I will blast through the code looking at general structures and entry/exit branches and the like, and note things and often breakpoint sections that seem hairy or where I can note inefficiencies or great design.
Then I slow down the stepping and really examine the heavy-lifting sections to try to become familiar with the style and abstractions being used.
This method has served me well, as more often then not, by the first day's afternoon I can be having intelligent conversations with the existing team, almost always much to their confused surprised.
I suppose it matters much if you are going into the senior position as really strong head-down coder or as really more of a project management liaison.
Not being much into the management side of things, I don't have much advice on that role, and your suggestions sound really smart.