What's a good level to start the Mines? I'm on Level 14 and I beat the caves but it takes me 7 hits to kill one robot o.O
Put simply, you need a better weapon.
At this point, if you like Saber-class weapons, try getting a Brand, and, if possible, a Buster. If you like handguns also, try for an Autogun, or a Lockgun. (though I doubt you'll find a Buster or a Lockgun. The store should start selling those to you at your level.)
It's normal to take 7-8 hits at that level, I would suggest using sword or partisan, since you are a HU, those robots are easy if you keep them on the floor, which is why area attacks of sword/partisan works well ^^
You are pretty good at PSO, I couldn't beat de rol as lv13 til I got my skill built up from my years of DC PSO training: I can easily beat normal ruins at lv18 HU now, with starter gears.