Original Japanese PSP 1000

They are identical, only some irrilevant hardware changes and modbo (barion takyon ecc, but there no very imprtant changes like the 200X series). I can make with DC 8 my psp 1004 like your model, no change
No, now you have a standard original psp 1000.
PSP support also WPA

 

Unless it contains a TA-079, its mainboard is found in all other regional models as well.

The PSP supports WPA as well, other than WEP.

 

Every PSP model/firmware to date can be downgraded/upgraded, with some restrictions:
- some models can't be downgraded below the firmware they came with (for instance PSP GO).
- on later models it's not possible to create or use a pandora battery.
- cfw can be installed on all models, lcfw is a users choice, unbricking a PSP Go is very difficult, therefore it's adviced to install lcfw on that device

Go for 6.60 PRO if you have a Slim or Go model PSP, you can play WiFi Ad-Hoc games online with that firmware, check this.

You can install a simple mod so you can boot your lcfw by clicking the Digital Comics (not available on JPN devices) or SensMe™ channels

If you have a Hackable PSP (all 01g and 02g models except TA-88v3) you can install DC Mod (VResurrection_Manager) and with it install/uninstall any CFW and any OFW you wish.

One more tip, you might think it has limited storage capacity? Wrong.
With the CR-5400 PhotoFast MS Pro Duo adapter you can use up to 64GB (GigaByte!) storage capacity, so plenty of room for your games, homebrew, music, videos and all other stuff...

It's a bit out of the picture but it's still a great device, and they're dumped atm so really cheap to buy one.