Hello shillamus,
First I would recommend you not limit yourself to only things that Amazon sells even if it's convenient. Convenience doesn't mean best for the job, or even best selection. Second, I'm not sure what air
pump your talking about because I didn't see a link to it, but it sounds like overkill for 5 gallons of water. One small twin output Aquarium pump for 30-60 gallon fish tanks is all you need. I get them at Wal mart for about $12-$15. Wal mart sells two brands (at least here they do), one costs about twice as much but claims to be whisper quiet. It's my experience that the one that claims to be whisper quiet isn't really noticeably more quiet. At least not enough to justify costing twice as much money. So I always get the standard one now. I attached some pictures of one I used in a 5 gallon bucket to grow a pepper plant.
As for the air pump getting hot and heating up the
nutrient solution. I don't have any experience with the large multi port manifold air pumps, but in my experience the smaller pumps I use like the one in the images wont change your nutrient solution temperatures. Some people have said they feel it was heating up their nutrient solution, but I haven't experienced that myself. In fact I even used that same principal to try and cool the nutrient solution. If warm air would heat up the water, it would stand to reason cool air will cool it.
So I did an experiment. I took about a gallon of room temperature water and took temperature readings before I started. I stuck the air pump in the freezer (the air couldn't get much colder than that) and ran the two air lines and plug out the door and closed the freezer. Then I plugged the air pimp into an extension cord. I let the air pump run for nearly an hour and the water temp didn't change one degree. If it couldn't cool one gallon of water at room temperature down one degree after an hour of pumping freezing air, it wasn't going to cool a 20 gallon reservoir either.
On the other end of the spectrum I have used the air pumps outside in temperatures over 100 degrees (including the pepper plant in the pictures) and didn't have any problems. I didn't do any controlled testing, but have never had an issue that caused a concern. Ambient air temperatures and radiant heat outside during summer months will naturally heat up the water. So insulating the reservoir and having to cool the water is normal anyway.
The pepper plant in the pictures, I warped the bucket with bubble wrap insulation and eventually set the bucket on a sheet of 1 inch thick Styrofoam insulation to block heat from radiating up from the ground. I would add about 1/2 gallon of ice every night and the water temps would stay comfortably in the low 60's all day. You would think with constantly pumping 100+ degree air into the bucket the water would heat up quickly, but it's never been an issue for me. I even did the same thing using an ice chest as the reservoir, as well as a 32 gallon trash can. I wrapped the trash can with 4 layers of bubble wrap insulation and again placed it on a 1 inch sheet of Styrofoam. I also made a lid using Styrofoam a total of 4 inches thick (four 1 inch thick pieces of two different size circles glued together) to fit perfectly. The smaller circles fit snugly inside the trash can, and the larger ones were 2 inches wider than the trash can to make a lip like a lid. I would add 4-5 gallons of ice to 30 gallons of water, and it wold stay cold (low 70's or lower) for 3-5 days.
As for
ebb and flow and alternate system designs. I don't know what your trying to grow so it's hard to give specific custom advice on generic information. I would have to have specifics on what your trying to do and accomplish before I can give advice on building a system to grow them in. I would be happy to help if you wanted me to.