A very well thought out post! I hope I can provide a helpful response...
In my experience, it seems that the best way to improve visualization is to simply "do tactics". Of course in your example it is not too dificult to find ...Qd1+, however, the ability to fully see this position before it arises on the board is, of course, more difficult. I would simply recommend focusing the majority of your time, whatever percentage that may be, to doing as many tactic puzzles as you can on tactic finder. (Also, don't get caught up in the time or make the first move until you are sure you see the entire puzzle because this will defeat the purpose! :D)
Other than this, one visualization exercise I have come across is to, without any board place a knight on a random square and name another square. In your mind, try to find the shortest method to arrive at the second square.
I'm not sure that was of any help...
P.S. Focus on the forcing moves first!
I use a 2.5" x 2.5" paper chess board that I cut out from a score sheet. I'll look at the paper and move an imaginary knight around the board, for example: find Nf5 [Ng1-Nf3-Nh4-Nf5], find Nf6 [Ng1-Nf3-Nh2-Ng4-Nf6] or practice openings, checkmates, tactics...everything and anything.
I find it much easier to calculate variations without pieces on the board. While playing internet chess, it's difficult for me to imagine my White dark square bishop capturing Black's knight on f6, look at the board, and NOT see that knight anymore, because it is STILL there. Well, it's not completely that bad, but it gets much worse when there are multiple captures, and needing to see the resulting position after those multiple captures.
So, I entered "chess visualization skills" into google, and the search engine filled in the blanks - "how do you improve chess visualization skills". I then stumbled onto a site I toyed with a year or so ago, www.chess-strategies-tactics.com. I did the sample lesson, which was not easy for me, and it was only a level two exercise. Here it is.
You are told to visualize 1)...Qxd5,2)Rxg7 Rxg7 3)Bf6, and then told to visualize the winning move for Black.
I couldn't do it, even after staring at this puzzle for 5 minutes. I just couldn't see the position clear enough that far ahead. Once I moved the pieces in Chess Base, I saw the winning move imeadiately, so it wasn't because I couldn't see the tactic.
How Do You Practice Chess Visualization? When did you develop decent visualization skills? After how many years? Was it something that just sort of happened as you grew up with the game over many years, starting as a child?
I started playing guitar at age 7, and I can visualize my fingers on the strings just fine. I can practice new scales, chords, anything - in my head - and then play it right the first time. I don't need to physically practice the guitar anymore to learn something new. I couldn't do this when I was 7, but at 53, and after 1000's of hours practicing the guitar physically, it's not too hard to see it in my head, and then play it.