Ignore the HSA. You can spend it for medical expenses even if you are no longer eligible to make contributions.
A "rollover IRA" (because you terminated employment and rolled over a 401k or 403b into an IRA) is treated for tax purposes like any other IRA. There is nothing special about it.
So, you now have an IRA with what can only be pre-tax funds, because the funds were originally contributed tax-free to a workplace plan.
Also, note that IRA means "individual retirement arrangement". You only have one arrangement, no matter how many different accounts at different brokers you might have. And since all IRAs are combined for tax purposes, you can make contributions to this IRA or to another account at the same or different custodian, it makes no difference.
What does make a very great difference is whether you can take a tax deduction for your contributions. That depends on your income and other situations, see this page https://www.irs.gov/retirement-plans/ira-deduction-limits
If you can take a tax deduction, then any contributions you make are tax deductible, all your IRA funds are pre-tax, and all future withdrawals are taxable. If you can't take a tax deduction, then you create an after-tax basis in your IRA. That leads to complications when converting or withdrawing, as you must follow the pro rata rule. You will get a form 8606 if you file a tax return with non-deductible contributions, that establishes your after-tax basis, and you need to keep that form to refer to it in any future year when you make additional non-deductible contributions, conversions, or withdrawals.
