1. What is rich of peak and lean of peak?
As mixture is leaned from full rich, the exhaust gas temperature (EGT) will rise until the mixture reaches the "perfect" (stoichiometric) combustion mix, and then will fall again after reaching that peak EGT. Any setting richer than that to give peak EGT is "rich of peak" and anything on the lean side is "lean of peak." These are abbreviated ROP and LOP.
2. On run-up when it's hot the article says to lean the mixture, do you so by setting RPM to 2500 or at the normal run-up RPM?
Neither. If the density altitude is high enough to lean for best power for takeoff (3000-5000 depending on the aircraft/engine combination), the engine manuals tell you to run up to full throttle and then lean. With a fixed pitch prop, you lean to peak RPM. With a constant speed prop, lean to the recommended fuel flow for that density altitude. If you lean at normal run-up RPM, you'll be leaning too much and will have less than best power at full throttle.
3. Following the previous questions, on the run-up on hot days it says 'After you're done with the runup, push the mixture lever forward "a bunch"'. Why? don't you get the best performance when it's leaned?
Ignore that. If you lean at full throttle as the engine manufacturers recommend, you'll be right where you want to be. Of course, if you lean at normal run-up RPM, you'll be too lean and will have to enrich to get best power, but there is no good way to know just how much to move the mixture control. That's why the engine manufacturers recommend leaning at full throttle, just like you'll have when you take off.
4. Richer mixture cools your engine better? why?
That actually depends on where you start enriching. Peak cylinder head temperature (CHT) typically occurs when the mixture is set so the EGT is about 75-100 degrees rich of peak. If you're lean of that point, then enriching will increase CHT, but if you're rich of that point, enriching will reduce CHT.
5. What are your thoughts on descending do you just run full rich?
That's the worst choice. As you reduce power with a carbureted engine, the mixture will go richer all by itself. If you started with the mixture properly set for cruise, reduce power, and then enrich, you'll go way too rich. The first thing that does is create an environment highly conducive to plug fouling, which means when you add power later you might not have it to add. Second, if you start at a high/hot enough position, the engine might go so overrich that it goes rough, and you'll have to lean more to keep it running smoothly. My personal technique is to just leave it where it was in cruise until I have to advance the throttle again.
All these issues are addressed reliably in your engine Operator's Manual as well as the Lycoming Key Reprints and Lycoming Service Letters 185, 192, and 197, which you can find on the Internet or obtain from Lycoming. Similar pubs are available from TCM for Continental engines.