Have you looked at Section 11.1 page 34? It specifically states the steps for reattempting a failed mission. I'm certainly not an expert but I will attempt to answer...
1. Offensive missions: Shall I put PC markers again on all cleared and not occupied cards?
no, but you leave the ones that were there from the first attempt.
2. Patrol missions: as above
[editted]. Not sure on this one, but I think Eric is right (you do place PC markers for each combat patrol)
3. What about enemy units? Would they be on map only for the first round of reattempted mission and leave after that according to Activity Chart (not under fire and no LOS to American units)?
11.1 states that you leave any enemy unit that is under Cover in that cover, but place any enemy unit not under cover under the best cover available on its card. Note that all enemy paralyzed, litter, and casualties are removed from the cards
4. Can I reposition my units freely assuming they will go to secured cards? In other words - can I mark secured cards, take away my units from map and put them again on any secured card in any order I wish?
Yes. You can also put them under existing cover markers, if any.
5. Do I start a mission with activity marker on "Contact/Engaged" side up if I reposition my units that they will have enemy units in their LOS and automatically open fire on them just as mission will start? Do enemy units automatically open fire on my units in such situation assuming they have LOS and are in range?
I think yes to both questions.
I'm really liking this game, but it certainly takes some effort to get it all straight (worth it though).