1) The gods are exceedingly active in my homebrew setting, both in an active role as persons and through the machinations of their respective cults, which collectively have a larger day to day role in the every day affairs of society than say the Catholic Church in medieval society. My inspirations here are decidedly Etruscan or Hindu in origin. This is polytheism to the max, because it's logical that in a society with deities known to be actively intervening regularly, the society would be more religious than almost every historical real world society. The entire world is as religious as a real world fanatical cult.
2) My general statement to the PC's is that divine intervention is so common, that pretty much everyone can expect some sort of miraculous event to occur in their lives even if they are but a humble peasant farmer. Thus, pretty much everyone attributes all coincidences to the supernatural, whether this is reasonable or not. Further, as 'heroes' they are very much heroes in the Greek idea of the word, in that their heroic nature is defined by the fact that the gods have taken a special interest in their lives. Therefore, they should expect divine intervention to happen pretty frequently. In fact, there is a roll they are allowed to make when they get into trouble, which involves rolling an 18 or higher on 3d6. I saw, "Or higher", because pious characters have a positive modifier on the roll as can other circumstances. Likewise, there are a variety of ways of improving your chances. For example, if the characters spend a destiny point, then they need only roll 18 or higher on 4d6. So far, we've had 4 appeals for divine aid receive direct intervention by a deity over the course of a 7 year campaign where the characters hit 10th level. There are rules likewise for determining just how strongly a deity intervenes - three of the four interventions involved the deity casting a 1st level spell (at 30th level of ability, but still a 1st level spell) on the players behalf, and one involved a 5th level spell (Monster Summoning V). Three of the four interventions were successful at saving the person from mortal peril, with the fourth (a cloud of obscuring mists intend to provide cover for the player's escape from overwhelming odds) narrowly failing owing to a combination of not taking full advantage of that and bad luck.
We've also had one bonafide god show up (and hurl a hurricane at the players), one literal demigod (though not a demigod by the rules, the offspring of a god), and several divine rank 0 power characters with reoccurring roles in the game - namely "Jasper the Talking Cat", whom they've gradually come to realize is more than an 'ordinary talking cat' as the full range of what he can do becomes obvious. We've also had the players receive a variety of dreams and omens from their patron deities warning them of upcoming troubles or steering them toward certain paths (for example, informing the champion where to find the steed his deity intended for his use). Normally, the players are appropriately careful when around beings of unknown power, though the one time a player seriously annoyed Jasper by being disrepectful, he reached over and 'scratched' him for 18 points of damage. The player who got scratched briefly saw Jasper turn from a house cat into a snarling black sabertooth tiger the size of a draft horse. The player shut up, and let the more diplomatic players do the talking.
Further, that's just the big time 'born from the fruit of the tree of life' deities that made humanity and the other moral races. There isn't a completely clear dividing line between gods and not gods. Divinity goes from the big family of deities that demand and expect worship (but can't compel it), all the way down to the ordinary tree and pond spirits. The fey are often referred to as 'the small gods', and are propitiated to a certain extent accordingly. And three of the allowed PC races are fey, so technically the party has a 'god' in it, albeit a very minor one, but the PC fey's prowess is justified in game by the fact that they are a small god, of domestic affairs and in this case domestic vengeance. (Actually, the current party has not less than 3 semi-divine beings, but the party isn't actually aware of that, and further one of the other characters in the party is a legitimate Saint, having had a conversation with his patron deity. And actually, the party has 3 saints in it, but they aren't actually aware of that either.)
3) As a player, I haven't had direct contact with a deity.
4) No, quite the contrary. I try to bring to my player's attention this aspect of my homebrew world as soon as possible, with some plan for bringing the PC's into contact with a god in the first or first few adventures. In the case of the current campaign, that didn't actually happen because the players leapt one way instead of another, and no even divine plan survives contact with heroes. But while deities have more reason to maneuver 'back rank' pieces about, they also have a lot of reason to maneuver the 'pawns'.
As for the complaint that the gods force mortals to do this or that, one of the major themes of the game is that they aren't allowed to. The gods are observing a truce in the aftermath of a war, and they aren't allowed to force mortals to do things against their will. Likewise, they are careful not to intervene because every 'move' that they make in the game that they play allows an opposing deity to make an equally powerful counter move. So, for example, this is why the deities when directly intervening the affairs of the party mostly only cast 1st level spells. Every time a good deity does that, some evil deity somewhere cast a 1st level spell on behalf of someone appealing to them! (Although, in fairness, one of the PC's was appealing to an evil deity - they'd made an acceptable blood sacrifice to him on his sacred ground and straight up rolled an 18 on 3d6, what else do you think should happen...)
Campaigns or stories with equivalent levels of divine intervention to my homebrew would include 'Chronicles of the Dragon Lance', 'The Lord of the Rings', 'The Iliad and The Odyssey', or 'The Order of the Stick'.
