A while back, I got completely locked away from my old steam account because of a hacker that bothered me more than a couple times. He completely changed all the information on the account so I can’t get it back. I was going to re-purchase a game that was already downloaded on my computer from my last account (I can’t play it since it’s linked to the other account). My question is, if I purchase it again on another steam account, will I need to redownload it or will I be able to use the already downloaded file to play?
No, you won't have to re-download the game files. From Steam's support page on Steam Account Use :
Can I use multiple Steam accounts on one computer?
Yes, you can access different Steam accounts from one computer. However, you may only access one account at a time.
The accounts will share the core files for any games the accounts have in common, but each account will maintain its own custom content and configuration files.
To answer the original problem, Steam does not use any of your personal information to restore control of your account. When I had to regain control of my Steam account, they asked for one of the Credit Card number used to purchase the games in the account.
Contact steam and retrieve your account.
Yes - See @Arkive's answer
However steam on my family's Mac has a steam folder for every different Mac user account, meaning you did have to install the game for every user account. I am a Windows / Linux person, so I am not 100% certain if this was correct behavior for a Mac. Luckily, I've found 2 solutions to this problem:
The easy way
In steam's preferences you can select a folder to hold game data. Create a folder in the shared user, and set it's permissions to read and write for everyone (RClick folder --> Get info --> Unlock button --> Permissions --> Everyone --> read & write). Then in the steam client go to steam --> settings --> downloads --> steam library folders. Add the folder you created, and right click it and make it the default folder. This will cause new games to install into that folder.
You will need to do this for every Mac and Steam user combo, but each game will only need to be installed once.
The easy way part 2
To move existing games from the user steam folder to your system-wide game folder, right click the game --> Properties --> Local Files --> Move.
The hard way
Creating hard links from all the user steam accounts to a shared steam account. It works shockingly well. You will need to do this once per Mac account, and then everything works.
I spent a couple hours figuring this out, so hopefully people in the future will be able to Google this question and get an answer.
Do you need to re-download it? Not Necessarily...
Will you be able to use the files you have downloaded? Yes! You can!
As of my experience, since the internet in Asia is horrible, I have been copying steam games from my friend, ("technically, it's illegal, but if you have own the game, no one can bother you")
Then I pasted it to my steam library (Ex: "C:/Program Files(x86)/Steam/SteamApp/Common") and let steam do the rest. (Depending where your SteamLibrary is, the location may vary)
You wont be able to directly play the game, since steam will verify and download few files, but at least you wont have to download the whole game from scratch.
Another way is using Create & Restore backup, but since you don't have access to you account, it wont matter anyways, but if you have, you wont have to download it from scratch. Its the same either way.
As other answers already mentioned it, no, you won't have to re-download games, although I just wanted to add something to Arkive's answer, more specifically the "but each account will maintain its own custom content and configuration files" part.
The catch is that some games do not specify correctly their save files and configuration files, and some games may get their data overwritten when another user logs in and synchronizes his Steam Cloud data. That happened to Dark Souls III notably with characters being overwritten when another user logs into his steam account and launches the game.
In short, back up your save files manually before starting any game, just in case.