Sell the bill? Market is very liquid.
I've never understood this mentality. I've got a credit card with nearly 6 months of expenses as a limit, most of it not used because I pay it off month to month.
I'd be more worried about the tax implications of selling ETFs and such than the 2-3 days of wait to transfer.
Maybe I lack imagination, but I can't think of anything that I'd need my entire emergency fund immediately. Throw the emergency cost onto a credit card, and transfer the money from the emergency fund to pay off the bill. I guess that doesn't work in some other countries where credit cards aren't as prolific as the US.
This depends on the bank. If you have a checking account with the same bank, the funds can be available right away. The standard withdrawal frequency limits still apply.
I'm not sure I've ever had a nontrivial expense in the US that simultaneously couldn't be paid on credit and wasn't known for weeks in advance either. I would expect any no-credit charge in excess of about a dinner bill to be announced at least a week in advance, implicitly through an invoicing process at the very least, and I would frankly have reservations about whether any vendor who didn't have a way to handle this was really a legitimate business.
Even if that's not an option, your broker is probably fine with floating you a sum of cash matching your recent MMF sale amount as overdraft or margin at a price of somewhere between "we've already factored it into our fees as a cost of doing business" and <0.05%/day of the loan amount anyway, because overnight loan alchemy is a well-understood problem and the bread and butter of retail banking.