I play on an HTC One, so with a small screen compared to a tablet. Here's the things that I find cause gameplay issues.

1) Trying to select a room to upgrade and not an individual within that room, whether zoomed slightly in or all the way out continuously selects an individual person.

2) Touching the screen to move around and zooming in and out, if not careful selects an individual, several times have accidently moved a person from 1 room to another. Should make it that you need to select a person before you move him/her.

3) While working on people's weaps/clothes and have others in barracks the cam gets pulled to the barracks to see them running to the back to "get busy", and even when the cam gets pulled to a room incident. It's annoying, I don't need to see it. Maybe just an alarm to alert that there's an incident or the Pipboy giving a thumbs up to indicate the other, then I can scroll out and find the room myself.

4) When people are revived from death (and i've had a couple rooms die before) characters go on coffee breaks instead of remaining in that room. Not an issue if it's just 1 person, but multiples from different rooms becomes a problem (in most cases that'll happen after a deathclaw attack)

Otherwise the game seems to run well on my phone, hasn't crashed, no lags or glitches that I have found.

First tap dweller, then drag? Nice idea. Seems to be both simple and effective solution for both problems above.

When dweller is selected, upgrade button for his room is available (for tablets). Well, installed FOS on smartphone to check it up. Indeed, upgrade button isn't shown. Though there is place for it.. I think, if idea "tap dweller, then drag" will be implemented, zooming in / out will be less painfull. Necessarity to zoom in to select room would be less annoying.

Discussed many times on forum. Showing "busy" pairs isn't necessary at all. Centering screen on incident without zooming in will be enough. Or zooming should be restricted to device's screen width.

Bug is in mechanics of returning dwellers to their work after incidents. Revived dwellers will return to their places if being revived after attackers are defeated.

I can relate to most of this. There are many times when I will just be moving around the screen and accidentally move a dweller to another room. So I had to resort to creating breaks in my play area. I use blank areas with nothing in it to tap and move around just so I don't accidentally move someone. The idea of tapping and then dragging works.

And taking away the centering in on couples getting busy is really annoying and needs to be changed. I mean I know they are because I put them there in the first place, doesn't mean I need to watch the room while i'm trying to get resources. Incidents that are caused by rushing should not get this mechanic because, well I caused the incident. Random incidents maybe but not rushed incidents.

and the selecting individual persons instead of room thing has happened a lot for me lately after the update. Not sure if they changed something within the code or not.