I'll give my opinion on this. The OP will be in regular print I'll be in italics
1. They said the trucks would run better after that. Is this true?
No. I don't know what else to say that is simply false.
2. What exactly do they mean by "run better"? Is it smoother, less noisy, or just have more power?
A smog pump, being belt driven does take some energy to turn, but if you take our belt off of it and go for a drive you'll realize it's insignificant.
3. Can I just cut the belt on the smog pump, or is that a bad idea?
Sure, you can do that, and if you want to try running without the smog pump please just try removing the belt first before you scrap the entire system
4. Will the truck smell terrible at idle if I remove all the smog stuff?
Probably. The main thing is that the smog pump helps the catalytic converter to work by introducing air / oxygen to help the catalytic converter to burn up the bad gases. Without the air pump it'll take the catalytic converter forever to get up to working temp, and in cooler weather it might not ever get up to temp when cruising. Even if you have already removed your cat, the air pump still helps to clean up the exhaust a bit. Have you ever heard of an exhaust gas called NOx? It's not as bad for the environment as other gases, so people don't talk about it as much, but it is REALLY good at destroying lung tissue in any animal that breaths enough of it. Any mechanic that has COPD is probably aware of this gas. Anyhow I'm not trying to be scary or anything like that, just trying to state the facts as I know them.
5. Will the check engine light come on? Any other considerations like that?
Maybe, but there is no downstream O2 sensor so I think it's unlikely.