What I've done:

-- I've upgrade /a, /ap, /x from 10.0 to 10.2

And what are the problems:

1) I can't run X. Whether i'm in root-mode or not i can't.

#startx
#unknown program in line <number>:xauth
#unknown program in line <nimber>:xinit

2) It's something wrong with PATH. I can run only the programs from /bin and /sbin. If I want to run any other one I MUST enter full path! But ->

#su
#export $PATH
#/usr/local/sbin:
/usr/sbin:
/sbin:
/usr/local/bin:
/usr/bin:/bin:
/usr/X11R6/bin:
/usr/games:
/opt/www/htdig/bin:
/usr/lib/java/bin:
/usr/lib/java/jre/bin:
/usr/lib/qt/bin:
/usr/share/texmf/bin

What is happening with the system?

What is your output when you do :
but logged as your user, not root

Did you follow the official upgrade guide?

Slackware 10.1 to 10.2 Upgrade HOWTO

I forget. When 4th runlevel is used everything is allright if i'm root. But if i'm user nothing happens and there are not anything in logs!