Yes, I recalc skills, HP, and so on when stats change - I generally track it but it's helpful if the players do it, too, to check my work.
Yes, I encourage players to use their own observation and reasoning skills in-game instead of relying solely on die rolls - I do try to work the character skills into the results however.
For example, for Spot checks, if the player says something along the lines of, "My character is looking around," it's a straight Spot check, but if the player says, "My character is watching the steeple of the church," then the DC for detecting the sniper in the steeple drops by 5 while the DC for spotting the machine gun nest in the storefront goes up by 5.
In another example, if the players miss an important clue that I present, I'll call for a general knowledge check to see if someone remembers a pertinent detail, then I'll pass along a clue-to-the-clue, to hopefully get them looking in the right direction.
I'm admittedly a grumpy grognard, so I recall well the days when as a player if I didn't tell the GM I was checking the ceiling, the floor, the four walls, behind the door, &c., then the character didn't either. While I don't advocate a return to those days, I do think that the players should be encouraged to use their ingenuity and be rewarded for interacting with the environment. As Thanee cleverly remarked, this isn't a boardgame, and as a player I don't want to passively sit back and wait for the dice to tell me everything that happens - I want to be an active participant in the events of the game, with the dice resolving the element of chance.