Phone is fine but I find if printed its quicker. Annoying standing waiting while someone digs out their phone then goes hunting for the required info. Some are organised some take ages
landing in Bkk the walk depends on the gate and if its S1 can be from few minutes walk to 15 minutes walk to immigration. Then domestic is 5 minutes walk from immigration
If your transit then you just go to gate for phuket. If you booked a domestic to Phuket then you go through immigration at bkk grab your bags and go to domestic.
Hopefully your on one ticket with Thai airways and your bags go direct you go to new gate and do immigration in Phuket.