Right now my current employer's 403(b) is in Lincoln Financial. If I roll it over to an IRA, what financial company should I use? Does it matter?
Find somewhere that lets you invest in low-cost index funds with no transaction fees. Vanguard and Fidelity are great options. Others are fine too.
And is the back-door roth only eligible for people who are above the income limits for a regular roth contribution?
Yes. Technically anyone can do it, but if your income is low enough to make direct contributions you might as well do it that way because it's easier.
What do you mean by zero pre-tax IRA balance?
The backdoor Roth IRA is a two-step maneuver. First, contribute after-tax dollars to a traditional IRA. Second, convert your traditional IRA to Roth.
On the second step there's a pro-rata rule that says that when you convert your traditional IRA to Roth, the conversion will be split proportionally between pre-tax and after-tax balances. You'll then have to pay tax on the pre-tax part, which you probably didn't want to do. The way around this is to avoid having any pre-tax money in an IRA at all, by instead leaving it in a workplace retirement account such as a 403(b) or 401(k).
Since one of my employer's account's is an after-tax contribution account with employer match, am I supposed to convert this to a roth since it is already taxed (other than the match)?
Converting after-tax traditional balances to Roth is often a good idea, because you'll be taxed on the gains if you leave them in a traditional account, but once it moves to a Roth account there are no more taxes on gains going forward.