1.Yes, given that you do not have to pick your bags and re-check them with United (and if you do, they won't open their counters until 4 hours before the flight because Newark is their only flight of the day, so consider the idea of using a storage service. There used to be one next to domestic arrivals, you can check the airport's website). If you leave the airport at 1.30 p.m. (get a taxi hired in one of the kiosks inside the terminal, or use the airport bus –private company, also inside–), you could be in Miraflores or the centre of Lima by 2.30, do things until 7.30, and arrive at the airport by 9.00. Two hours is enough to do everything and be at the gate on time. If you have checked bags, 3 hours are recommended.
2. To my knowledge LIM does not offer airside connections in most domestic to international flights for passengers like in all US airports. In most cases you have to get out of the domestic side of the terminal, get your BP, or if you have it, proceed to security and passport control.
Coming from CUZ you might be lucky enough and board as an international passenger from there, which means you do passport control and customs in CUZ, not LIM, and are routed directly from the national side to the international side by the airport staff. If this is the case, no visit to Lima, I am afraid. Check with AV that this is the case before doing plans... In any case, the TUUA is already included in the tickets. So, if you don't pass passport control in CUZ, you should be able to leave normally and re-enter in the international terminal whenever you decide to do it.
3. All passengers are made equal in Peru. No express security line –or anything close to Clear or TSA pre-checked. It is hardly needed. No shoes out, no laptops out. The only things banned are liquids bigger than 100 ml, which I guess is around 3 US oz. Maximum time I've spent in a line was about 5 minutes –and the line was moving all the time– and from the point you put your stuff in the rollers of the scanning machine to the time you pass by the magnetic detector –also, no 3D scans in Peru– can be around 30 seconds.
4. Also, once you pass security you can buy all the (overpriced) liquids you want, but they will be taken from you upon boarding because all US-bound flights have a pre-boarding security check by TSA-trained local security.