I'm only level 54 on my Sam B, but I have yet to play the Bloodbath Arena.
I am essentially ready to begin Act 4 on Sam B, and this is only my second play-through. I have 50/50'd the original game and have yet to start any of the 10 DLC achievements.
That context aside, most of my experience has come from completing EVERYTHING and noting challenges that I should complete. There is no shortcut to getting to level 60, the best methods I've found is strategically using Fury against special mobs (like the Floater Swamp near the Hospital) or aiming to complete challenges.
As for your question about better XP in second play-through, I can not CONFIRM this but I would assume this is untrue. I would say the figures are larger because as you level up the quest reward is set to give say 1% of XP, so as the requirement to level becomes higher so too does the figure (despite it still being 1%).
The turn ins are a really beneficial way to stack up either money or XP, though money is quite useless outside of the consistent repairs you must do.
Finally back up your save files and that should eliminate playing in fear, I haven't personally experienced it and a lot of people have said the patch amended it but it never hurts to be safe.