Well, the first thing that can be said here is that your router is faulty.
There is no way that a new wifi DSL router should lose the configs on a soft reset (which is just powering off and back on). The only thing that will happen is you normally get a new IP, which is transparent to you anyway.
The same thing applies to an 'auto-DMZ' which is what you seem to be describing. To place an IP in the DMZ must require an admin change, it cannot possibly occur by chance - or as a default
So it's simple:
1. Return the router, it's a dud.
2. If it's 2nd-hand or something - bin it and get a new one. That's why your 'pal' gave it to you, it's kaput.
Normally you must do a hard reset to lose the configs, wipe them, and revert to factory defaults. The procedure for this varies but is basically:
1. Power down.
2. Hit the reset button (hidden on the back panel).
3. Hold the reset button in and power up.
4. Keep the button in for 20 seconds.
5. Release it.
Now you should be back to factory defaults - no admin password, no ISP settings.
Get another router, yours is foobar.
ps --- You'll have some fun and games if you try to set up a DSL router for cable broadband, or a cable router for phone line broadband. The cable router has no modem in it. You'll get some way down the track but it won't work. However the faults you describe aren't related to this, at least, every time I screwed up badly enough to be doing this, I never saw any of those ones