Can I travel to Europe from the US and back to the US on a Mexican passport?

First, you can apply for an expedited passport, as noted in comments. Even regular service is quite prompt. The last time I renewed my passport was nearly ten years ago, but I did not take the expedited service and it arrived in just over two weeks nonetheless. The passport service is fairly prompt, though things may be different now, I suppose.

If you will be returning to the US before your US passport expires then you might want to postpone getting a new passport. You can travel to Europe with your Mexican passport. The Schengen area requires your passport to be valid for at least three months after your anticipated departure from the Schengen area. Your US passport does not meet this requirement, but your Mexican passport does (or so I assume).

You must, however, return to the US using your US passport. A US citizen can use a US passport to fly to the US as long as it is valid.

If you do this, you would show your US passport only two or three times:

• When you check in for the flight to Europe, show your Mexican passport. If they ask about your immigration status in the US, show the US passport as well, but point out that you'll be using the Mexican passport in Europe because of the validity limit.

• When you check in for your flight to the US

• When you arrive in the US

Everywhere else, that is, to European immigration officers, show your Mexican passport.

Would I be denied entry upon return to the US? Would I have to get a visa? Even if I am a U.S citizen and resident?

You cannot fly to the US using (only) your Mexican passport, because no airline will allow you to board the plane with that passport unless it has a US visa in it. But you cannot get a US visa because you are a US citizen.

Having said that, you cannot be denied entry to the US because you are a US citizen. If you were to find yourself in Europe after your US passport expired, you could do one of a few things, including:

1. Fly to Canada or Mexico with your Mexican passport and then apply for entry at the land border using your expired US passport and any other evidence of US citizenship you might have.

2. Apply for a US passport at the nearest US embassy or consulate.

3. Wait for your new passport to arrive at home and have someone send it to you.

From EEA (Europe) website (link below0 "There are thirty-nine countries, including Mexico, whose citizens do not require a visa to visit the EU for a period of up to three months." The same applies to the UK. Remember to bring evidence that you'll be leaving Europe at the end of your trip, such as return ticket. Be able to demonstrate that you have access to sufficient funds to support yourself. Bring evidence of medical insurance/travel insurance. Have some accommodation booked and if staying with friends/family write down their addresses and phone numbers, 'cos some airport wifi signals are poor and it's tricky if you can't easily provide this info to the Immigration Officer.
Link https://eeas.europa.eu/delegations/mexico/14899/travelling-and-studying-eu-mexico_en