I have a tendensie to switch players from position to position based on their stats. Some players have low position familarity without any blue markings. Will their familiarity to that position grow or stay the same?

It will grow.

I like to keep very deep depth chart at all positions so eventually I will have a lot of players familiar with positions. This way in case of injury or multiple injuries I will hopefully have someone who can fill in with some familiarity.

For example: All my O-lineman are in depth chart at all O-Line positions, All LB's are in all LB positions as well, Safeties as CB and so on.

That's an interesting idea. Do they grow in position experience by being in the depth chart for that position, or do they actually have to play? Or both?

They have to actually play. They gain experience between sims only at the position that they are officially assigned to (with no regard to where they are in the depth chart) but each player gains experience for each play they are on the field at any given position.

Yeah but you never know with injuries, I have had my whole O-Line get blown up in games before and without the deep depth chart it will start pulling players like FB or TE who may not be as skilled to play. I also pretty much balance my weights for those positions so most of my O-Line rates the same at all positions because honestly your RT better be able to pass block as well as the LT because teams will put the best pass rusher at the LDE position

Since I read somewhere that players won't get any position experience from Special Teams during the training camp (maybe I'm wrong), is it a bad idea then to play my rookies on STs during training camp?

If you want players to develop experience at ST you need to play them in ST positions. It's not a bad idea at all.