Thanks for sharing the link to those chess problems. It reminds me of a paperback book I picked up last year called Bobby Fischer Teaches Chess. It's full of problems like that, taking the reader from beginning backrank mates to more complex combinations. It's a very fun book to read, especially the second half, as you hold the book upside down.
The thing I've found is that chess problems, and the Bobby Fischer book, only helped my end game, and the planning, if I remember to do that. It is the opening that can make or break a game, and I've had many games broken by a bad opening.
I'm experimenting with lightening games lately, and while I lose almost all of them on time, I find it is a good way to practice openings, and to force my brain to work faster.
Maybe I'll see you on freechess.org. I have the same handle there. I'm still very much an amateur.