#floating mana

[magicjudge]

Can I float mana in my main phase, to use in my battle phase?

No.

Mana empties from mana pools as each step and phase ends. You cannot float mana from one step or phase into another.

[magicjudge]

If my opponent controls a card like Propaganda, when can I activate non-mana abilities (e.g. Deadeye Navigator + Palinchron) and float mana to get enough mana to attack with a bunch of creatures? Or is there no time frame for that?

There’s no time to do that.

Declaring attackers is the first thing that happens in the declare attackers step, and players don’t get priority in the step beforehand. Mana can’t float between steps and phases without cards like Kruphix specifically saying otherwise.

[magicjudge]

I have 7 forests in play. I cast Summoner's Pact, get Omnath, Locus of Mana, and cast him. At the end of my opponent's turn I float GGGG. I untap and draw a card. If my opponent notices now that I didn't pay for pact, what is the procedure? If they waited for me to cast a spell, then pointed it out?

It’s fairly clear here that you paid the pact trigger here. You announced that you were floating the mana, then you didn’t announce any floating mana when you moved to your draw step, so the pact was paid for. 

While you should make a confirmation like “pay for pact”, it’s not always done. No one calls judge when a player taps four lands in upkeep and then draws without saying they’re paying. There’s nothing to suggest to me that the trigger was missed here.