Removing links from the chain does not correct the issue of the chain being stretched. It may solve the rubbing issue but that, depending on just what is rubbing, may be that the freewheel (FW) is simply too big for that particular rear derailleur (RD)
Going back to your original post, I personally find the practice of shopping for cheap parts online and then asking a shop to install them a bit tacky. If you want to have a good relationship with the shop let them recommend the parts and buy them. You don't shop online for car parts and ask your mechanic to install them do you?
If the chain is stretched and judging by the looks of the brake levers I would not be surprised if this bike does not need a full overhaul. You may not need new cables but I bet the bike would really benefit from some new grease in the bearings and wheels being trued.
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“One morning you wake up, the girl is gone, the bikes are gone, all that's left behind is a pair of old tires and a tube of tubular glue, all squeezed out"
Sugar "Kane" Kowalczyk