Hmm. You've got a lot of questions there, and I'll try to answer them. FYI, I'm using a TicWatch Pro, Tags, and have a 'Link' as well.
I don't know if a Samsung watch is any different, but with my watch, simply tilting my wrist wakes it up, and it automatically displays the GolfPad F-M-B distances for the green. I'm not sure if you have some Bluetooth setting on your watch where it times-out after a few minutes of inactivity (and the you have to reconnect each time, which is causing a delay). You don't want that setting that requires a double press of the Home button to open GolfPad. GolfPad should own your watch when you're playing a round.
As for putt tracking, I'm doing this, and this is where Tags are beneficial, especially on the green. I understand that GPS will never be accurate enough to track putts that accurately, so that's why I usually mark pin locations while others are pitching, putting, etc. and/or I adjust my putt locations after my ball is in the hole and others are putting out. That way, I'm not holding anyone up.
GolfPad will not normally register sand shots, punch-outs/recoveries, and can make errors on fairways due to GPS accuracy. But you can adjust any of these shots by selecting the shot from the list, and selecting the ellipsis on the right side of the desired shot, and selecting 'Set lie'. I do some of these things in a quick 2 minute review post-round.
For statistics, there are some differences between the phone app, and what you find on the website. To see a 2D map of shot dispersion, use your phone app stats, go to 'Clubs', select a club, and then from the subsequent screens, swipe to the right until you see the 'Accuracy' screen.