Is there a way to mod a psp 3000 and still be able to fully turn it off without it deleting the mod?

If you can get Wi-Fi on another network
(library, school, or a friend's house), then it's either the router or your PSP's Wi-Fi hardware standard. An 802.11g device needs an 802.
11g router, basically. Hope this helps.
It's very straight forward. Put your memory stick in your PSP
(if it's the first time you're using the memory stick you should go to the settings on the PSP and format it - Mac DVD ripper could rip DVD to AVI, multiform videos/audios Rip DVD movie to video and audio formats including AVI, MPEG, WMV, MP4, DivX, H.264/AVC, MOV, 3GP, 3GP2, MP3, WAV, WMA, AC3, M4A, OGG, RA, AU, etc.
this will make all the necessary folders)
then plug your PSP into the PS3 using a USB cable.
Go to the PS1 game, press triangle and select copy.
It will then copy over to the PSP. DVD MP4 Converter for MAC provides fast access to high-qualified, easily handle DVD to MP4 conversion.
Game saves can also be copied over -
go to the virtual memory card on the PS3, find the save and select copy then choose PSP.
1. You can not use the PSP browser or the PSP Flash Player to watch youtube videos online.
2. If you want to get into homebrew/CFW you can try PSPTube.
3. You can convert Youtube Videos to MP4s using sites like vixy.net with your PSP or PC.
Or you can get PC software converters.
4. The Flash Player for the PSP is fairly limited. If you want some Flash games try
psponme.com
5. For direct video downloads to your PSP check out pspube.com
Try Linksys or Belkin.
Their routers are better.
Check the hardware on the PSP, as well.
You should have a router that is compatible to your PSP's standard Wi-Fi.
Check the standard of your PSP.
If your PSP has 802.11g, don't buy an 802.11N or 802.11b standard router.
There's a BIG difference in the standards.

M2TS converter is a multifunctional software which helps you convert m2ts files to AVI, MPG, WMV.

It may be that your PSP may not be capable of Wi-Fi.
Check that, as well.