Yes, they've let me bring large-ish purses in. They ask that you not bring umbrellas in unless you keep them in a bag or purse, but they have a place where you can check the umbrellas, like a coat check, if you need it.
Yes I usually have a satchel with me in the museum.
And camera bags (they encourage non-flash pics. And art bags (good spots to sketch).
Yes
Yes you can.
Yes I just left there and yes several people carry bags
Yes, you're fine with bags and backpacks.
Yes. My purse was about 16 by 12
Yes, at least the time I went.. a month ago.
Thanks! Your answer is awaiting moderation.