Yes, you file married filing jointly. You don't really have a choice anyway (except married filing separately, but that's rarely the right choice), but people like you are who that filing status is most advantageous for since you get to "use" the deductions and lower tax bracket space that your wife is not.
What matters in Modified AGI, not AGI, so you may need to add some things back in to line 38 of Form 1040 (see
https://www.irs.gov/publications/p590a#en_US_2017_publink1000230489 for what you need to add back in), but yes, as long as your MAGI is below $99k you can make a deductible traditional IRA contribution of $5500 regardless of whether you're covered by a retirement plan at work (it sounds like you weren't since neither you nor your employer contributed -- I know the "covered" term is confusing, I had it wrong for awhile until someone corrected me). In fact, as long as you have at least $11000 of income, your wife can also make her own $5500 deductible traditional IRA contribution as well!
For your general question, what matters is your MAGI which is AGI (line 38, form 1040) with some things like IRA deductions and student loan interest deductions (see the link above) added back in. So it doesn't matter what your "salary" is, just what you're actually paid (according to your W2) over the course of the year. Things like employer paid health insurance, 401k contributions, etc won't be included in your W2 Box 1, so they help reduce your AGI and your MAGI.
So if someone "makes" $120k/year (as in that's the stated salary), but only works 9 months for a total gross salary of $90k, and contributes $18k to his 401k (note that the limit for 2018 is $18.5k), then his AGI would be $90k - $18k = $72K. There would probably be other reductions like health insurance, but for simplicity we can ignore that. So his AGI is 72k, and since there have not yet been any deductions from the link above, his MAGI is also $72k. If he files MFJ and this is all the income in his household then he can make a full $5500 deductible IRA contribution, and his husband/wife can also make a full $5500 deductible IRA contribution. His AGI then becomes $72k - $11k = $61k (note that his MAGI is still $72k).
For 2018, if you and/or your employer is contributing to your 401k and your MAGI goes over $121k (see increased limits at the links
here), then you will no longer be eligible for deductible IRA contributions. What you'll want to do at that point is call your IRA custodian and ask them to "recharacterize" your 2018 traditional IRA contribution to a Roth IRA contribution. They will move your contribution and any gains on that contribution to a Roth IRA making it effectively as if you had contributed to a Roth IRA in the first place. You can recharacterize any time up until the tax filing deadline for the year. You'll need to attach a statement to your taxes explaining what you did.
You got the math a little off in that $18k + $5.5k = $23.5k, but otherwise you're right that that would take that amount off your taxable income. Note that the 2018 401k limit is $18,5k, and your wife can also contribute $5.5k to her IRA even if she doesn't have income (as long as you do).
At your income level you don't need to worry about the backdoor Roth for awhile. Just forget about it until your income is too high to make Roth contributions (MAGI of $189k in 2018, higher by the time you're there).
An E-fund is just shorthand for emergency fund: money you keep around in an easily accessible place (like a bank account) that's invested in a way that's not likely to lose value (so not stocks) in case something unexpected comes up that you need money for.
Fidelity is a good option (we have almost all our money there). You need to be a little more careful with them than Vanguard since they offer lots of not so great investment options along with lots of great ones. Yes, you want to signup for a Traditional IRA. Have your wife signup too if you can afford to lock away an extra $5500 (make sure you have enough of an emergency fund first).
I would suggest you invest in a target date fund as they offer a good mix of investments all in one fund so you don't have to worry about investing in multiple funds. With Fidelity you want one of their Fidelity Freedom Index funds (note that the "Index" part is important as they also have actively managed funds call Fidelity Freedom funds with higher expense ratios for no real benefit).