Can I run my previous Minecraft installation(s) with a new account?
Assuming you buy the same version (i.e. Java Edition), yes. You'll just need to log in with the new credentials. You may need to download the latest Launcher, but the actual Minecraft app should not need re-downloading.
Will I need to rename the dat files tn the playerdata folder with my new ID?
This question seems to suggest yes. If you don't know the IDs, you should be able to note what the existing file is, join with your new account, and see what new file was added. The new file should be your new ID and the other should be your old one.
Alternatively, there may be mods out there that allow you to access other player's inventories. For example, NBTExplorer
Is the format just "<id>.dat"?
From this Reddit post:
In single player your data is stored in level.dat - the external (UUID).dat player files are only used if someone else joins the single player world in LAN mode.
Do I need to do anything else to move my player data to a new account?
You will likely lose capes that are saved on Minecraft's servers. You may be able to contact support to get them transferred. Aside from those, if you had a uploaded skin, you may need to re-add it using this link
Are there any known bugs/issues that doing this may cause?
Based on the Reddit post above, there may be issues transferring XP, recipes, and advancements.
Is it the same with servers and modded installations?
Assuming you own the server, you should be able to adjust the files in the same way. If not, you'll need to talk to the server admins to see if they will transfer your player data. Knowing your old and new IDs would be helpful, if not necessary, for this.