So, I have a 2 1/2 year old android - Galaxy Samsung 7. It gets hot (in a non-predictable way), and it gets slow - usually when it’s hot.
A phone repair shop owner (where I was checking out for potential cases) said that the battery was going bad, and that if I got that replaced it would fix the problem. Is that reasonable? Is ~$45 reasonable for that fix?
Many smartphone batteries are only meant to be recharged 730 times (i.e., once a day for two years.) At 2.5 years, yours is already likely past its service life. And $45 is a bit on the high side, but still a fair charge for the replacement. Since these are Li-on batteries, they pose a serious danger if they were to overheat and ignite/combust (aside from causing a fire, they also could release hydrofluoric acid fume as part of the combustion process, which can be lethal. Unlikely and only a remote risk, but still not one worth risking.)
As Velocity said, 2 1/2 years is a pretty good run for a li-ion phone battery, and that phone is from a time that Samsung was pushing the state of the battery art pretty hard. (That’s a contemporary of the Samsung Note 7, also know as “Zippo”). 
The price is reasonable if you’re not already a DIYer, you don’t already have the tools, or your time is worth more to you than $40 an hour or so.
And yes, the battery getting discernibly warm during charging or use is a bad sign. The only three worse are, in order of severity: very poor charge holding (shorter run time off charger); perceptible battery swelling; and spontaneous combustion or smoke generation.
The phone can also get hot and slow if there is a process on it that is using lots of CPU for a long time. It is needlessly difficult on Android to figure out which, if any, app is consuming lots of CPU at the moment. You can go to the battery manager and somehow get to a list of the percentage of battery different apps are using. Screen, Android System, and Android OS should be at the top, but if there is some app that is using way more battery than it should, then that might be the source of the problem. Of course, if you do something like use Google Maps to navigate all day, then expect it will have used lots of battery.
The shop can’t charge you for removing a buggy app, but they can charge your for replacing a battery. That doesn’t mean the battery is good, though.
Another thing to check before getting the battery replaced is to see how much of its original capacity is available. I use the Phone Info app to do that. Under the Personal tab it should have a battery health percentage. My 1.5 year old S8 is at 98%. I’d say consider replacement if it is below 80% or so. I think that app will work on your S7.
Basically any smartphone will be noticeably warm if you run something on it that maxes out the CPU/radios.
An old battery will make your phone run out of charge faster and it could cause excess heat and slowness, but it’s not the first thing I’d look at if someone reported a warm/slow phone. That probably means that some misbehaving process in the background is churning through processor, which slows everything else down and heats everything up.
Cool app! My battery is 75%, definitely time for a new battery!
And my CPU temp is 44C. This is why it sometimes doubles as a handwarmer for me…
44 C is way too hot. It should ideally stay under 40, and best under 35.
My old Samsung Galaxy III would get extra warm at random intervals as well as having the charge draw down for no discernible reason. I figured some app was responsible but was never able to track down which one it was or why it was not more consistent. But than I was unable to figure out why the phone would have a short burst of vibration (but no sound) every morning at 9:32.
Maybe you had an alarm clock accidentally set to vibrate mode at a particular time.
Probably the battery. Also worth studying what apps were installed or updated recently. Occasionally rogue apps can cause high CPU and a hot phone.