The smaller diameter pipe is because it is specified for in-wall use. The socket end of a standard 3" PVC pipe coupling is 3.97" which is wider than a 2 x 4 stud wall cavity, which is 3.5". The 3.25" pipe OD on the in-wall pipe allows for a smaller coupling which can fit inside the wall cavity.
Depending on the size and type of leak, you might have been able to seal it up without having to cut it out. I put a nail for a picture frame through a drain line once. To repair it, I scraped some shavings from some scrap PVC, mixed it with a little bit of PVC solvent until it formed a paste and then filled the nail hole with it. Note, that I would never do this on a pressurized line.
I'll try to break down what the inspector is describing. Keep in mind that these inspection reports will show "defects" that are simply things that do not meet certain recommendations they look for. They may not be up to code, etc. The problem with some inspection reports is all defects are called out in the same way. One defect can be seriously dangerous while another on is merely an annoyance. Understanding why it is considered a defect is the most important thing.
You basically have two issues. The first is that the A/C condensate drain is tied into the temperature and pressure relief overflow for the water heater. Imagine the pressure and temperature in the heater rises to a dangerous level and the valve activates. That's going to force scalding hot water to flood your air conditioner because they are both connected. The A/C housing will leak and the PVC pipe will not stand up to the temperature or pressure for long. This is not the way it should be done. The A/C condensate drain should flow into a nearby drain pipe. It's common for it to tie into a bathroom sink drain or similar.
The second issue is a little less clear, but we can assume it's related to the first. The inspector is saying that there is not a proper overflow drain to the outside for the water heater. This could be because the overflow doesn't run all the way outside, but since the condensate drains there it probably does drain outside or into the sewer. He probably means the overflow drain isn't functional because of how its tied into the A/C.
So, if you have a gas water heater, and you don't have access to a proper drain, you can get a special gas valve like the Watts 210-5. This is a temperature sensitive valve that will cut off the gas supply to the water heater if it gets too hot. The idea is that cutting off the gas stops the dangerous heating of water without it getting to the point of blowing water out of the TPR valve that doesn't have a proper drain.
The Watts valve seems to be pretty expensive - online prices range from $250 to $400. On top of that you will need some additional gas piping, and if someone else is installing it, you could be in the $800 neighborhood pretty quickly. Also if the water heater is old then any time you mess around with the pipe fittings going into the tank, you have a chance of doing irreparable damage if they are corroded (have to replace the water heater).
If the copper overflow pipe does go outside, then rerouting the PVC A/C condensate drain could be cheaper.
Best Answer
"Suction" is not desirable.
The main problem there is likely inadequate slope on the line coming straight out from the air handler, and/or inadequate slope on the leg to the air gap. The line across the top front appears to have adequate slope.
The actual trap is beyond the air gap, in larger diameter pipe.
In my biased opinion, the second major likely cause of problem is the use of small diameter pipes and sharp elbows rather than larger pipe and DWV sweeps for drainage plumbing. HVAC types are all about "it's just clear condensate, I can run a tiny pipe" and then they are shocked, just shocked when it clogs up (because that condensate comes from air, that has dust, that includes spores, and things grow...)
The open pipe stub (on top of the furnace) is probably intended to allow you to clean the vertical pipe, but given there's no access to clean the horizontal pipes, that is of limited benefit.
If you are actually asking about the open pipe on the P trap entrance, that's quite specifically to prevent suction from draining the trap and allowing sewer gas into the house. It's a "standpipe" forming an "Air Gap" and is utterly required. The P trap itself appears to be correctly installed to a drain line with a vent line extending upwards, though we can't tell if that drain line or vent line are clear and functioning. If you pour water into the standpipe it should drain away quickly, but water should remain in the lower part of the trap when you stop pouring water in, not be sucked out into the drain.
Post-update Ugh. Well that was a sloppy hack job, as revealed by your latest/last picture - the vent line is not complete, and is venting sewer gas into your house. If there was an inspection of this work, the inspector should be fired and anything they passed should be re-inspected, and whoever did the work should be publicly shamed as well as being made to fix it to code. Or perhaps taken to court to force them to pay for someone actually competent to fix it to code. That open vent pipe inside the house on the smelly side of the trap is certainly the source of your sewer gas smell, and a code violation the size of a gigantic tortoise held up by 4 elephants...
It should either extend all the way through the roof, or connect to another vent line that goes all the way through the roof. Some might suggest a "Studor" vent also known as an AAV (air admittance valve.) I would not be among the "some" - those things fail, and your house stinks again, and you replace them, and they fail again - a vent through the roof works, reliably. Perhaps as a very temporary, quick stop-gap while you get the vent to the roof sorted.