Will lock picking a door in plain sight affect relations with a town or npc

Just figured this out. You're not supposed to steal from the case on his stand. Go behind it and look down. There is a lock you can pick behind and under the stand and his strongbox is inside this area.

I've had success with a few things when it comes to people following me. I'm not on a sneaky character, so I've had to get crafty for some missions. It sounds like you've tried some of these methods already, but I'll put them here anyway just for completion's sake.

One is to try during the day, when people are out and about or tending to their stores. I've noticed that some (not all) vendors will stay at their counters instead of following me.

Second, is to go in and scout a quick path while letting the person follow you, but not steal anything. Then, go out and come back in (this usually resets their position) and make a mad dash for whatever you needed to get. I may just be lucky at encountering slow people, even in Solitude. I'm not sure if their speed or concern with me stealing something is related to anything.

Finally, I've also had success with telling my companion to wait on the stairs or in smaller door ways to block the person from following me, or at least slow them down more. It seems to be hit and miss, but it's worth a try.

I've also read (and it's mentioned in a comment above) that buckets/pots on heads can work. I've never tried it myself, but it would be a lot funnier than any of my suggestions! Maybe some combination of buckets and companion blocking to slow them down would help.