5 ounces is the standard or "generic" amount of sugar that comes with most kits, it produces around 2.6 volumes of co2. If you are just starting out that's fine to do. Other folks carb to style. But of you are starting out that fine.
Heck even though I often carb to style a good percentage of the time I just go with 5 ounces and don't bother with the chart, or it calculates out to that much anyway.
The kit makers know what they are doing.
As to letting it cool, opinions vary since you are dumping it into a bottling bucket, which should have some sanitizer in the bottom, and you are racking cooler liquid into it, it's going to drop temps quiclkly.
You can if you want, let the priming solution cool. It's really a matter of preference.
I have removing solution from stove rolled into my sanitzation routine, so it ends up cooling off for about 5-7 minutes whil3 I'm sanitizing the final case of bottles, and gettig ready to rack.
But dumping it straight off from the stove won't hurt anything either, like so much in this hobby there are many ways to do things and they all work fine.
I outline my process and offer lots of tips here in the sticky above.
https://www.homebrewtalk.com/f35/bottling-tips-homebrewer-94812/