Yes PS1 memory cards work in PS2 systems BUT and this is a big but only for PS1 games not for PS2 games. Because the file systems are different PS1 writes data in blocks whereas PS2 writes in bytes.

PS2 games require a PS2 multitap, but can often (but not always) use PSX Dual Shock controllers, however there won't be any analog support for the buttons (the analog sticks should work just ...