Do they have lockers during events?

I did not see lockers, however there is a coat and baggage check on each side of the building.

Yes

Yes, McCormick place provides locker as well as coat and baggage check in each side of the building.

Do they have gender neutral bathrooms?

McCormick Place does not provide lockers. Depending on your event, there may be coat and baggage check available for a per-item fee. Prices and location can vary depending on the show organizer. They do not accept laptops, cell phones, furs and purses.

Yes, on the street level

Yes I believe it's at the bottom level

Yes they do, on the bottom level

Yes. On the left side.