Does this game have it? Previous games did and it caused lots of problems.

:D

Saves do grow over time, but I think it is still the Autosave system that is the main culprit for save issues. Reason, the autosaves are created at Cell Load/Render which means the cell is sometimes incomplete when the save is created.

I have never had a save related problem in any TES game or Fallout 3/NV game, but that is because I never use the Autosave features, I never load an Autosave, and I save either before I enter a new cell or only in the middle of a cell but never as it loads or shortly after.

I also never overwrite a save slot more than 10 to 15 times, once I have that many overwrites, I create a new slot. I have also noticed that sometimes a save takes a long time to write, and when that happens, I wait a couple of seconds, then save again in a new slot.

I haven't seen or heard that problem yet but it could be possible. Me personally I usually delete auto saves if I switch to a new character. I may take CCNA's advice and create a new save every so often, I too have been noticing a lag in save time in my one file, at least around 5 seconds longer to save. I blame it on the large file size for the memory (15MB's, Xbox 360).

This.
Also, sometimes it is advantageous to reset all game cells by waiting.
Just travel to an unimportant interior cell and wait.
TES4's reset was 72 hours whilst TES5's is 720. That equals 30 in-game days.
And, as it was best to wait 73hrs in OB to be sure, I find it best to wait 31 days in Skyrim.
Your Save-game size will reduce.