An IRA is an investment account, a HYSA is a savings account. You’d be hard pressed to even meet inflation in a HYSA, let alone make any real earnings. You put your emergency fund in a HYSA because you don’t want it to be invested. The goal of an emergency fund is not to make money, but to have access to a few month’s expenses in case of an emergency like losing your job. You do want to invest your retirement savings, as you can make a good chunk of money by letting it grow in the market for a few decades. Having that money sit in a HYSA is a ton of missed potential.

You can withdraw from a HYSA any time. It’s just a decent savings account at a bank or credit union.

You can also move some of your emergency fund into I Bonds, which are guaranteed to match inflation, but you cannot touch those for a year. You will want to make sure you do not lock up your entire fund for that period. It’s common to do something like move 10% of your fund into I Bonds each year. I would keep some in a HYSA too just since that’s easier to cash out in an emergency.

Store your emergency fund in a HYSA, yes. You can pull money out of the Roth IRA any time, but earnings are subject to taxes and penalties if you withdraw before you’re 59.5 years old. There are some exceptions to that, such as first time house purchases, but generally it’s just a tax-advantaged retirement savings account.

I haven’t looked too hard into why the roth ira is recommended before increasing 401k contributions, since I’m not there myself yet.

Generally the order is:

• emergency fund (3-6 months of expenses)
• get any 401k match your employer offers
• pay off high interest debt (roughly 5% or more)
• max out Roth IRA
• increase 401k contribution until you’re contributing at least 15% of your income to retirement
• save for longer term goals (like that house)

You can see the prime directive in the subreddit wiki for more info.