Can't access internal storage from normal system but I can in TWRP? I have Lenovo K3 Note and I flash NEXUS-V2.5 rom. System works fine I think but I can't access anything through file managers,sdcard, sdcard0, sdcard1 are empty, but they have files on them, which are available when I boot in TWRP. TWRP version is 2.8.6.0. When I try to push files from adb I get error Read-only file system. adb works fine in recovery.
Try to execute following commands via ADB if you are rooted (in what I strongly doubt):

adb shell
su
mount -o remount,rw /system


or

adb shell su -c "mount -o remount,rw /system"


After that your push should work.
If you are not rooted, you will fail on the second command. If so firstly you should acquire root on your phone.

What concerning your question about customs ROMs/recovery: most custom ROMs already contain root patches. However, flashing TWRP has nothing to do with root and is done separately.