the_3rdplayer: Interesting. Might be cool to finally test this properly for the first time and find out what the true requirements are!
This is how I know for years:

finish under 5 hours -> infinite magnum
finish under 3 hours -> infinite rocketlauncher
finish under 3 hours -> infinite minimi for chris, infinite ingram for Jill
finish with all characters survive -> special key for costume change room
it does not matter how many times you save for any of this.
it does not matter how many health sprays you use for any of this.
it does not matter if you use special weapons (minimi,ingram,infmagnum) for any of this

I have to admit tho, that I always finish the game around 3 hours, sometimes more sometimes less. so i usually get everything. but the rocketlauncher is interesting to me. I'm pretty sure it does not matter how many times you save but you need the good ending. Might be my memory is wrong and it DOES matter how much you save.

Shalazah: After experimenting with RE1, here's what I've found about unlocking the bonus items on this version.

You get the Closet Key for getting the best ending.

For finishing the game in under 4 hours as Jill, you get the Ingram weapon (an infinite SMG). For finishing the game in under 4 hours as Chris, you get the Minimi weapon (an infinite light machine gun).

To get the infinite Rocket Launcher, you have to finish the game in under 3.5 to 4 hours without saving. You're allowed to use infinite items like the Ingram or Minimi to do this if it's your 2nd playthrough onwards.

aureilia: The Rocket Launcher supposedly can be unlocked in any time as long as you don't save according to https://www.youtube.com/watch?v=3Cfnl6Bt8_o About 31 minutes in, his info is based on the 90's PC port which is what this is.

the_3rdplayer: I also heard from someone else that you can get the rocketlauncher by either time or amount of saves.
So either you finish under 4 hours with 0 saves, or finish under 3 hours with 3 saves. I finished under 2 hours with 1 save and didnt get it. So last one is a lie! Only the savecount is important, everything else doesn't apply for the rocket launcher.
But maybe it applies to the other two bonus weapons. More testing required.

Homekill: Beat the game and no matter what you will get the machine gun unique to your chosen character.
Time doesn't matter for anything in this version.

Homekill: It's actually very simple.

Beat the game and no matter what you will get the machine gun unique to your chosen character.

Beat the game with the best ending to get the closet key.

Beat the game once as Jill and load your completed file to play on hard mode.

Beat the game without saving to get the rocket launcher.

Time doesn't matter for anything in this version.