I've never seen a 386 which doesn't support a 1.44 directly in the bios (though if you try and use an 8-bit ISA controller that may not...) (-though it must have been possible since the 80386 originally came out in 1985 and the 3.5" HD in 1986...but that'd be a real early 386 board (ie 12-16mhz systems, likely with no ability to be upgraded)
Zip Drives require drivers, CDRoms/DVDRoms require both a driver and MSCDEX to assign a drive letter. (there are some 'generic' driver packages which work with just about any drives)
early 486s and earlier (down to the 286) typically only support up to 524MB drives directly (a drive overlay such as Ontrack can be used to access larger drives) (this is a problem of inconsistencies between the bios int13h spec and the ATA/IDE spec, limiting drives to 1024 cylinders, 16 heads, 63 sectors, 1024x16x63x512 = 528,482,304)