Turning off when your battery is at 5-10% sounds like a sensor calibration problem. Just plug it in, charge to full, leave it charging for two more hours, then unplug and use normally until the battery is drained and it turns off automatically. Then charge it up to full +2 again. That should recalibrate your battery sensor so that it will give accurate readings for remaining charge.
As for your second problem, that is strange. The first thing I would say is
don't steal apps. Installing cracked apps is not only wrong, it can mess up your phone. Second, many Cydia packages have not yet been updated to work with 4.0, and many of those can cause problems with your phone. Winterboard in particular can put your phone in a recovery loop. If you would like to know which apps are known to be working, check out
this spreadsheet. Be careful what you install, or you might end up in a boot loop you won't be able to get out of without restoring.
ETA: No 3G has a tethered jailbreak.
Yes i read somwhere that all IPHONE 3G are untethered, i just ask to be sure, cause here are many good experts, and with more quality answers than on another places

... I doubt that this iBoot version is maybe problematic...
Thanks very much for your explanation and suggestions! I really do not know why this is happening because I never had similar experiences like this now!
Maybe you're right that the sensor calibration cause, because it is sometimes strange when iPhone is connected to a computer, iphone shows 5% battery provides full, even though the iPhone fill 30-40 minutes, but after a period when i connect - Disconnect the iPhone, battery status changes to 90% .. I do not really know why the iPhone 3G itself off, I do not know what could cause this?
I did not actually install the application too, just sync app for OS 4.0 from source cydia.hackulo.us, I needed it to install igo my way 2009 ... Tnx for this google. Doc! I'll take this in my mind...
I do not know what's with this version iBoot, whether it can make a problem with the jailbreak redsnow?
I tried 2-3x restore OS 4.0 and then jailbreak again with redsnow-last version, Unlocking the ultrsnow later, but again the problem with switching off, this happens independent, sometimes, not just after 5, 10 or 15 minutes, sometimes whole night phone is switched on, and only tomorrow day it switch off from unknown reasons!
If you have some experience with the iPhone and itself switching off, post here...
Strange thing is that i have always to CONNECT IPHONE TO PC, to be able power on again, this seems like on tethered iphones, even iphone 3g is't tethered... This is ver dificult, to connect always iphone 3g to pc , after rebooting...
Thanks for your help and sugesstions!
Greeting