One hour 10 min layover in Newark; International -> Domestic connection. Enough time to clear customs?

Even without immigration and an infant it would be very tight on most airports.
The time immigration takes does depend on the time of day and day of the week but also on your relative location in the queue/line and with a baby everything takes more time.
I have been in an USA immigration queue for more than an hour with it being just one plane, different airport.

Customs can take no more time than you need to walk through one short corridor or it can take quite a while if they want to check your luggage and there are already people waiting. An hour wait is realistic.

If your ticket is for both flights you can risk it, knowing that the airline will get you to your destination at the first possible option. Also because many flights do land early.
But if you have important things to do shortly after arrival or an ongoing connection not covered by the same ticket or travel on separate tickets I would not even risk it with three hours.

You have added that you and the baby are both US citizens and the class of the single ticket. That is good, as you will have the easier line through immigration, (which you may be able to improve on if you can get one of the faster methods to get through the checks like Global Entry TSA pre-check,) but you will still have quite a distance to walk and a baby to take care of. A sling or carry frame for your child may make it easier, as it is rather likely you will not be able to use a pram, (due to it not traveling in the cabin with you.)

So I see your chances as slightly better than the average passenger on your flight, but it can still be a tight fit if the flight is on time and impossible if your incoming flight is late.
So it is still rather likely that you miss the plane and will have to wait in the airport for the next one with space. You may like it better to book the later ongoing flight (so you know you have a few hours in the airport and be sure you get on the next flight) and maybe invest in a hotel room (so your little one and you can get rest in private) or in buying entry for a lounge. (I have never done that but I often read here that it is possible and worth it.)

That's really quite tight and not something I'd be comfortable with, especially with checked bags and an infant (I might give it a shot with Global Entry, no checked bags, and no infant, but that's not your situation here).

The minimum connection time is about 70 minutes, so you're right around the minimum the airline will even let you book. Keep in mind that you need to be at the gate at least 15 minutes before departure time, further reducing your time. There may be enough padding on your flight from Barcelona that you can gain a little time if it arrives early, but there's no guarantee of that.

Connections like these at Newark often require that you change terminals after immigration and customs (it's possible that both flights will use Terminal C), which means going up some escalators, riding the AirTrain, going down some escalators, and then through security. That will take up even more of your time. (I've also experienced atrociously slow baggage service at United's international arrivals at Newark, though it's been about 18 months.)

You can use the CBP AWT site to see past wait times at immigration for the same time and day of the week, which may help you make an informed decision.

I'm not going to say it's impossible if all goes well, but I think there's a decent risk of missing this connection, and any one thing going wrong (delayed arrival into Newark, long lines at immigration, long lines at security, etc...) would be enough to miss it. I'd look at what your options are if this is the case. If you've booked this as a single ticket, United will rebook you on the next available flight with space, but if that flight isn't until tomorrow (or the flights are full), you could be in for a long wait, with your infant, without your checked bags.

If you do decide to try it, and you're a US or Canadian Citizen, I'd get the Mobile Passport Control app, which could help you get through immigration more quickly.

Based on my experience (including missing a flight, and getting stuck in an airport for an extra night without a visa to go out) anything less than 2 hours, that includes customs and border control involves risk of missing a connection and going through rebooking of the flights. There are many reasons why it may happen, that include suddenly longer lines, random custom inspections, passport checks, and many other ones. Unless you target for establishing a world record for getting the least layover time for a flight ever, you may consider to plan for calm, more relaxed route, to have time to grab something at the airport or visit a restroom without a need to sprinting it.

As a bonus point, you may find maps of airports and really plan your route, including transferring between terminals and international/domestic zones. I believe that this is the one for Newark, and it may give you a better view on the transitions https://www.newarkairport.com/at-airport/airport-maps

Oh heck no.

The problem is, you have so many steps there, and a complication. It might have a chance if you were OJ Simpson (once most famous for moving through airports very quickly), but with checked bags and a baby on board, this is risky. You would have to a) do everything fast, b) do everything just right, and c) have everything out of your control go just right - no late plane, long immigration lines, slow baggage pickup, or extended scrutiny at Immigration.

The stress amplifies all sorts of problems: the rush increases chance of injury. It greatly increases the chance of mistake, because you're flustered, and don't have time to double-check and ask. And it stresses you out in the Immigration and Customs line, which can make that go wrong.

If you booked independent tickets, cancel the Rochester segment and rebook something realistic.

If these are all on one ticket, the airline is responsible for the connection. Move consciously with no wasted time, but refuse to rush. If the airline sends someone to whisk you through Immigration, baggage collection, Customs and TSA, as well they ought... then take full advantage of it, of course.

Figure on a singificantly delayed arrival in Rochester, and set your hosts' expectations accordingly.

Even for me this would be tight and I have a NEXUS card, often flying business and a ton of experience moving through airports and not a baby in sight. The reason I am writing this up because people do not emphasize the baby enough. She/he will be very fussy after landing, the landing hurts their little ears. Hat tip: yawn at that them, they will copy and hopefully pop their ears. Nonetheless you are landing at around 8pm Barcelona time and haul them through a smorgasboard of noise, colors and smells at a time when they would be dead asleep at home. Unless you have a saint, they won't welcome that at all. There are nursing suites (type nursing suites and click search) at Newark, most definitely schedule some quiet time even if you don't nurse them there and then. Yes, you can rush through the airport trying to make this connection but a) you won't make it b) even if you do, the next flight will be unpleasant. Or not. Baby might just decide to fall asleep and stay asleep. I mean, babies :) but that's not an outcome I would expect.