There is no requirement for your employer to provide the HSA. Frequently the employer can get better deals on the HSA than you can individually (like no waived maintenance fees and the like) but you can for sure open your own.
In fact once you have enough money in there that you are moving out of liquid investments then you definitely want to shop around for HSA much like you would for an investment account.
Can you actually fund a non-employer HSA with tax-deductible funds, or is that a benefit that only applies to payroll HSA contributions? If not, a lot of the benefit of an employer HSA disappears.
(FWIW, I have an HDHP, employer HSA, and an individual HSA that I roll my employer funds to annually for lower-fee investing.)
Only partially. If you fund your HSA outside of your company's payroll system you can deduct your contribution amount from your MAGI for income tax purposes, but you still have to pay FICA taxes (social security and medicare) on that income.
If you have your HSA contributions made through your company's payroll department, then you don't have to pay FICA taxes on it.
HSA Authority (part of Old National Bank). Last time I ran the numbers, the break-even point on annual expenses vs some other options was about $10k in HSA balance — where smaller balances would pay more at HSAA than some other provider(s). But they cost less than other providers for larger balances (again, last time I checked).
Yes you can, go to Fidelity and sign up for one. Takes like 10 minutes. Transfer money, buy a fund, away you go. Just like an IRA, except never taxed at any point. Besides employer matching it's the best freaking retirement deal there is.