As long as the batteries are putting out the exact same voltage, then yes you can use a battery with a higher ma rating.
Nope.
no
Yes it will give you longer use time between charges.
Yes, it's safe I ordered a PSP battery from there
Yes. As long as the voltage is the same which it is. The 800mA is the capacity that the transformer can produce safely without going into an overload state. Your original adapter was rated at 500mA which means that what ever device was plugged into it draws less that 500mA. You might notice that the new adapter is slightly physically larger. So you are safe to use the new adapter with the higher rating.
no
no
If there is a hole through your battery, do not use it.no
Yes, but it is not as safe as a sealed battery.
Twelve volt battery chargers are just another battery you carry with you or keep at home. They are just as safe as any other battery you use. Just like any battery, they will have battery acid within them and have the potential to leak, but chances are very slim if you care for it properly.
heck no
No, of course not.