No. romw.bin is a different BIOS dump from the original SCPH-1001 model. The 5501 is a later revision with minor bug fixes. Most modern emulators prefer the 550x series.
Critical note: Emulators are case-sensitive on Linux and macOS. The filename must be exactly scph5501.bin (all lowercase). psx scph5501.bin
The SCPH-1001 (original 1995 US model) has a bug in the CD-ROM handling that some homebrew games rely on. SCPH-5501 fixed that bug. For retail games, either works, but 5501 is more reliable. The BIOS provides the standard file system for
Even newer emulators like Mednafen rely on a BIOS dump. The only exception is PCSX-Redux, which can operate in a BIOS-less HLE mode, but game compatibility suffers. but game compatibility suffers.
The BIOS provides the standard file system for memory cards (the mcwd:/ interface). Different BIOS versions have subtle differences in how they format and read saves. Using the wrong BIOS can corrupt saves.
The only legal method is to dump it from your own physical PlayStation console. This is akin to making a backup of a game you own.
Here is the legal principle (in the US, under fair use jurisprudence and the Sony Corp. of America v. Connectix Corp. ruling):