Oops, for some reason that dupe showed up as a dupe but wasn't a dupe
ANYWAYS
Correct, you can just put the PSTV in standby mode and Henkaku will stay. When you reboot, it'll have to be reinstalled again. You won't need to reinstall any homebrew or games already installed.
For PS1 games, you have two options. You can use a PSP mode eCFW to run PS1 games, or you can use native emulation via Retroarch's PCSX-rearmed core.
For PSP mode, there are two eCFWs, ARK and Adrenaline. ARK is an XMB-less (so no regular PSP GUI) eCFW that supports PSP games and homebrew in full, and PS1 game support but without sound. Adrenaline is a full fledged eCFW with XMB (the normal PSP's GUI), will support full PS1 games in the future. At this time we don't know when the update to add support will release, but we know it'll come eventually.
Native emulation, as I mentioned, is done via Retroarch's PCSX-rearmed core. Since it's native emulation, and doesn't rely on the Vita's actual PS1-emu, it doesn't run all games at 100%. Most 3D games will run between 80%-100% speed depending on how demanding the game is, and most 2D games run at around 100%.
If you had to choose either option, at the moment ARK would probably be the "best" way to play PS1 games despite it's lack of sound.
If you need a tutorial on installing ARK or Adrenaline, you can find them here around the Temp.