Okay, so, in order:
1)Change initial materia, level, gil, items and stats.
WallMarket handles this in its 'Initial Data' section
2)Change boss stats and their drop.
Proud Clod is the tool for this
3)Change/remove some treasure chests, materia, weapons and items.
Treasure chest contents are defined in field files; you need something like Meteor or Hack'7 to change them. To edit materia, weapons and the in-battle effects of items, use WallMarket. To handle the out-of-battle effects of items, you would have to disassemble the executable yourself.
4)Change level/stats limit.
The easiest method is to change the stat gains / stat growth / Exp. to level up in WallMarket. You could probably get rid of the 99 level limit with some disassembly, but there'd be little benefit after doing the above.
5)Change weapon stats and materia slots.
6)Change AP, required to level up material.
WallMarket, once more.
And a lot of different things.
Use White Choco for editing shop stock / prices, Enemy Manipulator and Hojo for global enemy stat changes (eg doubling Str of all creatures, giving all enemies immunity to Stop by default), my own Libre for editing limit breaks, Proud Clod for enemy attacks and AI, and my Disc Extension Patch if you intend to use slightly larger KERNEL.BIN and substantially larger SCENE.BIN files.
I want to mod PSX release of this game, because PC version sucks.
How can I do this and what program should I use?(P.S. Sorry for my English).
Yes, yes it does suck. That's why you need something like CDMage beta to handle your PSX ISOs - it makes it quite easy to reimport files into the ISO, which you can then play on an emulator / burn and play on a modified PSX console.