Your front derailleur inside limit screw isn't properly adjusted. Shift into the big cog in back and the granny gear. Set the inside limit to where it starts to scrape the chain then back it off a 1/4 turn.
As far as your chain climbing gears in the back. You need to adjust the cable tension. If that doesn't work, you may be experiencing problems from a slightly bent derailleur hanger.
If you adjust your cable tension and it works in lets say the bottom half of the gears, but hangs up in the top half, it's usually a bent hanger.
Your shop should have a derailleur alignment tool. Have them check it for you. If it's not too bent, they can usually bend it back. However, realize that a hanger is made of aluminum and doesn't hold up too well to repeated bending, rebending.
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