What is the proper way to maintain temperature in a water cooler Mash tun by adding hot water?

I agree with the tip that most of heat loss comes from the lid and opening it. I pushed a meat thermometer through the side of my tun near the drain valve this let's me see the temp without opening. If you don't trust the seal (hasn't failed me yet) you can buy a threaded bulkhead style one. I also throw a thick old blanket over the whole cooler and for extra insulation. I don't recommend the spray foam into the lid didn't seem to help and just made an ugly mess on mine. Blanket was much better!

I used a 10-gallon Home Depot cooler to build my mash tun, and I have no trouble with heat loss during a mash. The initial heat loss is due to the difference in temperature between the hot water and the cool grain and mash tun itself. After the temperature stabilizes (the cool parts have warmed up) the temperature should remain plenty stable.

You have to account for the initial heat loss by heating the strike water above the target mash temperature. Software can help a great deal with determining the temperature of your strike water. It varies, depending on the type of mash tun and the amount of grain. I use the free Brewtarget myself, but BeerSmith seems to be the most popular.

As for the leaky mash tun, I haven't encountered that. I ordered a cooler conversion kit from Northern Brewer, and it worked with no cooler modification required. I used two coolers and two of those kits, one each for mash tun and HLT. No leaks, no fuss.

As stated by others, it's easy enough to adjust your mash temperature by adding hot (really hot!) or cold water in small amounts until you hit your target. I've found it's much easier to err high and then adjust down with cold water than to try to have boiling hot water available.

This cooler should not give you any significant loss for a "normal" mash at ambient temps (~70F). This is exactly the kind I use and I typically see 1 degree or less lost over 75 minutes. When I do an overnight mash (8-12 hours), it loses ~30 degrees. Make sure you use the correct strike temp; add your water, then grains to water slowly while stirring and you should have no difficulty. Check the temp and adjust to reach your mash temp (see below) and then close the lid reasonably tight and don't open it until you're done.

Check the thermometer with boiling and freezing water in a different container. I have a cheap digital thermometer, and it's OK, but typically, I just use the dial thermometer I installed in my tun. As long as I look at it straight on, it's pretty good for reading accurately, and every few months when I fully clean the tun, I also check the calibration. 9 degrees sounds like a lot in 15 minutes, but then perhaps you had warmer and cooler spots. If you stir your mash properly, the heat should be evenly distributed. Stir, stir, stir as this will give you better efficiency and more even distribution of heat and help avoid any clumps of grain not getting mashed. Some people spring for a thermapen and they are nice; near immediate, accurate readings, but will set you back a c note.

If you miss your mash temp, add hot water to raise it, cold (or tap temp) to lower; start small and stir well. You should not have to adjust other than that for a 60-75 minute mash. I normally don't change it for a 90-120 minute mash, even though it loses 4 to 5 degrees by the time it's done. Preheating your cooler with consistently warm water (like your highest tap temp) will help even out your strike temps (because it's always hitting the cooler at the same temp) and will prevent you from cracking the cooler wall if it's really cold when you add the strike water (storing it inside will help avoid this).

OK, so @DennyConn and @brewchez's advice is very good, but they don't directly answer your questions. So here it goes.

I have found that the best thing to do is to keep a pot of boiling water and a pot of ice-cold water ready when I mash in. When I have adequately stirred my mash, I check the temp in 4-5 places, and go with the average. I add hot or cold water by estimate, a little at a time while stirring, to hit my mash temp. Generally, if you have a good cooler, you should not suffer more than a 2°F drop over the course of a one-hour mash. (I will note that there is strong evidence that a 30-minute mash is more than enough for one-step mashes, and maybe 15 is enough).

The key is to not open up the mash tun because the heat loss from that may exceed the benefit of checking (OK, I will usually do a check at 30 minutes). You can repeat that adjustment process if you want to adjust mash temp mid-mash.

You could do a bunch of math and figure out how much water to add and at what temp, but by the time you got you additional water measured and at the right temperature you are looking at a target that has moved away.

I will second @DennyConn's endorsement of rectangular coolers. I have excellent experience with the Coleman Xtreme line, but any of the 5-day coolers are great.

As far as your question, "Should I just chock this up as a loss and buy a different cooler with a 1" hole?": No. I don't think the hole size is the issue. if you have a silicone washer and a metal washer on each side of your cooler bulkhead, and thread the ball vale and the interior fitting on tight, then it will seal even if you hold is a jagged 1-1/4" hole.

On the other hand, should you "chock it up as a loss" because your specific cooler is not manufactured as well, or because it is a round beverage cooler and tends to hold heat not so well? I will leave it up to you. I would try wrapping it with blankets and/or reflectix insulation. I learned to hold mash temps in a two-gallon, $8 beverage cooler with terrible thermal mass, using only blankets, so all is not lost.

Most of the heat is usually lost through the lid in coolers. Cooler lids are not well insulated. The bodies are. This is because they are meant to keep things cold not hot. Heat rises and a cooler lid isn't designed to actually handle it. Some coolers are better than others. I have used several and found wide differences.
I found that if I covered the cooler lid with a couple old bath towels or a folded heavy wool blanket it held temp fine. Even for some extended mashes beyond 60 minutes. Some people drill some small holes into the lid and spray in some minimally expanding foam to insulate it.

Try preheating the cooler with some hot water first. A couple quarts of near boiling water sloshed around does a good job. More importantly, it heats the airspace inside the lid. Dump that water out repeat your test. But check it only at 60 minutes. In the ten gallon cooler, every time you open the lid you need to reheat a good portion of the air above your 5 gallons of water. Its nice to collect data, but the 60min point is the only one you care about.

Seal it, cover it check it in 60 minutes. I bet the results will be satisfactory.

Lastly, remember that for most mashes conversion is likely complete in 20 minutes anyway. So if you lose 2-4 degrees at 60 minutes your mash was pretty good at the start during the most active part of the mash.

I would definitely recommend a different cooler. I find rectangular coolers are much easier to use and I've never had trouble holding temp in one. I have 48, 70, and 152 qt. coolers and never lose more than 1-2F over the course of the mash.