What have I done??

You put it into recovery mode, just power it off by long-pressing the power button. Turn it on again and it should start normally.
How old is the battery? If it's the original (which would be amazing) or several years old it's probably shot and that would account for the faster discharging. Remove the battery and inspect it for any signs of swelling, a swollen battery is bad and should be replaced. The slowness could be a combination of a weak battery and the overall age of the phone.
You can look in the system menu to see what apps are using the most resources (power and memory) and try clearing cache for those apps or maybe even replacing them with different ones, but ultimately you're dealing with a device that's about 8 years old and outdated so there may not be much you can do. The S5 was a great phone in its day but that day has come and gone, I used mine for about 5 years but retired it over 3 years ago.

 

You hit the wrong button combination, I forget what it is to get into the mode where you can select the cache wipe.
Yes unfortunately OEM batteries are no longer available, and if you do happen to find any "New old stock" ones they will be bad from sitting around all these years. There should still be aftermarket ones available, just stick with the ones that are stock capacity and avoid the ones that claim to be high output as they generally aren't.
The big picture is that even with a new battery it's still an old phone with hardware limitations and can't be updated any more. You will have to go shopping for something newer.

 

No, you have the correct combination. When holding them, after you get the initial "Galaxy S5" screen, you need to let go of Power while continuing to hold the other 2. You'll probably see "RECOVERY BOOTING" in blue in the upper left - that's how you know you're on the right track

 

I agree with the forced obsolescence but that's just how it is with this stuff. My S5 has been sitting on the desk for the last few years as a paperweight since I replaced it with an S9, which itself is nearing its use-by date although currently still doing just fine. The thing I miss the most about the S5 (ok aside from the removable battery) is the IR blaster, I wish more phones had those.

 

I'm glad I ain't the only one who loves their old phone, although mine is an HTC Thunderbolt I got in 2020 via Amazon. There are ways to prevent unnecessary battery drain (I also got an S5 lying around).

Go through the apps list in settings, and check the little 'restrict background data' on each app. Turn master Sync off except for email and Google account, and NEVER use Wifi. For some reason, turning on Wifi on an S5 makes it run super hot and drain like mad. I think when the S5 was new, data plans weren't all unlimited so the TouchWiz interface kinda hard-coded a limit to background data usage.

Also, DO NOT update your apps. Newer versions not only eat up storage, but demand more CPU/RAM and background usage and kills battery faster. Also runs slower. Once I got back 2GB and uninstalled all the updates, the phone could manage a day or two of standby usage, and 3-4 hours of SoT.