Yes, if you have a modern Cobra CFW* (in rebug that's an option you enable with their optional install "rebug toolbox" app) you can just mount backups and run them from the XMB, no launcher needed (there even isn't one for PS1)
*HAN still doesn't support PS2 isos and you need a launcher, but that's not your case
That's normal, the PS2 emulators (ps2_emu, ps2_gxemu, ps2_netemo - you can only use netemu on your console) are separate kernels so the console reboots when you open or close them!
(If you used Webman, that would be the same reason you can't use FTP or software fan control in PS2 mode)
Yes (as do most backup managers on Cobra), but PS2 is an exception (in part for the reason above, modding PS2 mode is a lot harder and less flexible - being able to play isos from internal instead of using iso.bin.edat was a very significant achievement)
Yes, you need to undo any compression (ecm for data tracks, ape = monkey's audio for audio tracks), open the cue file in a text editor and check that the filenames for the tracks are correct, then open the cue in isobuster or cdmage and save everything as a single bin+cue - that's the only format supported for backups of streaming audio games
(actually some backup managers support audio with img+ccd too, but bin+cue is better supported)
Wait, you converted a PS1 game to eboot and then that to a PSP package? Didn't even know you could lol
That's just a name used for historical reasons - someone (probably Cobra when that name meant a paid CFW) first used it and everyone followed for compatibility
(it's even wrong - the console is officially called a "PS", unofficially "PS1" and codenamed "PS-X"; "PSX" is the media center version of the PS2)
You should be able to use ISO (worst format - can only contain a single data track), CUE+BIN (only a single BIN file due to the way the PS1 emulator works), and IMG+CCD at least!
Because not all games were tested and not everyone who did thought of adding them to ps3devwiki