Is that in order? I'd say having six months' expenses saved should be higher up in that list. After you've paid off all debt besides a mortgage, but before you start maxing out a 401k or IRA.
I'd put maxing out a Roth IRA ahead of saving for emergencies. You're limited to a certain amount you can contribute each year, and you can withdraw your contributed principal from a Roth with no penalty—it can effectively be your emergency fund (or part of it). Of course, long-term you definitely want a separate emergency fund to avoid needing to raid your Roth.
Hmmmm that could work, but I worry that if people treat their Roth IRA as their emergency budget in the event of a recession their total amount they can withdraw will be lower than their needs.
That is why keeping an emergency fund as a separate savings account at a bank makes me feel more comfortable. At most I'd put my emergency fund in a money market account to at least earn something, but the point of the emergency fund is to be liquid and available for emergencies. If there's a steep recession and all of a sudden you need to tap into your savings for 6-12 months until you find another job, you don't want to be doing that when your Roth goes down and cashing out your contributions means selling more shares than you purchased.