These problems are almost certainly not caused by Qt (except maybe indirectly like by filling up the disk), I have it installed on several occasions and never encountered such behaviour (it's not even used by default so can't really interfere with applets and hildon until you start a qt app).
/mnt/initfs (/dev/mtdblock4) 65% of 4M / (/dev/mtdblock4) 96% of 249.5M /media/mmc2 (/dev/mmcblk0p1) 10% of 1.9G /media/mmc1 (/dev/mmcblk1p1) 64% of 14.6G /debian (/dev/dm-0) 84% of 1.2G /debian/var/tmp (/dev/mtdblock4) 96% of 249.5M /debian/home/user (/dev/mtdblock4) 2% of 6M