1: Anything which is +X cards and +Y actions is immediately very promising, especially if either X or Y is greater than 1. I knew this immediately when I played with Alchemist. Seeing the same principle in action with Laboratory (effectively the same thing), Village, Market, and various other similar cards convinced me of its immense power. This is also what can keep turns going really long, as people will often have to shuffle their decks in the middle of all that drawing.
1a: If you have +1 card +1 action, you're good. You have effectively established the definition of "cantrip". Anything beyond that is merely you asking for some particular effect. In the case of Laboratory, that's increasing your hand size by 1 (-1, +2). In the case of Village, it's the ability to use more non + action cards, since + action cards already have that worked out. Village is definitely there for action-heavy decks, and in that respect, I can see why I overheard someone banning it.
1b: Now that I think of it, this seems to be an overall strategy by itself here, made for skilled players. It definitely has the most enjoyable experience to Timmy.
2: Never underestimate the value of money. It takes 8 coppers to buy a Province. That would take a lot of doing. Using silver, you're only required to have 4. Unlikely, but much more realistic. Once you put gold into the equation, you suddenly need only 3 cards: 2 golds + 1 gold or 1 silver. Copper is one you can underestimate just fine. In fact, that's the whole point of Moneylender. Once you have silver and gold well in hand, you can start ditching that little copper to get more spending power consistency out of your 5-card hands.
2b: There is clearly a whole separate strategy involving money. You don't necessarily want more cards or actions. The values you want increased are buy and money. Only upon writing this do I truly, truly understand why Festival is so good. It does work well with any strategy, but especially so with the money strategy.
3: You don't necessarily have to buy things if you can make them. This one I'm still gonna have to take some more time to understand, but it involves lots of "upgrade" type cards. Mine and Remodel are the big sticking points here. Getting Mine really early on (say from a particularly lucky 5-copper opening) will pay huge dividends. Remodel will let you turn 6-cost things into Provinces, and no matter what the spread, you are guaranteed to have one such card available to you in each game: Gold.
4: Stay in gear. While in "low gear," you need to get your strategy off the ground. Figure out your strategy and what it will take to start it. A bad start can ruin your entire game. All things being equal, you do not want victory point cards until late in the game. They contribute to nothing while the game is in progress and will slow your play when you draw them. Of course, Island is the exception to this rule in that you do want it early on. It'll allow you to get points while simultaneously removing your victory cards from your "playable" deck. The acceleration of the game will depend on how similar everyone's strategies are. If everyone's grabbing the same piles, you have to decide when it is most appropriate to shift into high gear and go for those points. Keep in mind, removal of 3 piles will end the game. If everyone's doing their own thing, you can probably stay in mid gear and work on those Provinces.
Card specific notes
At the moment, I'll talk only about the base set. Further analysis of the expansions will have to wait until I have the time to examine them.
Adventurer: To be honest, this one hasn't come up in our games, so I don't have a realistic opinion on it. Obviously it's there to game in the mid-game at best, because you're not going to afford 6 early on. I'm sure someone will hit upon its usefulness and I'll be embarrassed at my foolishness. In the meantime, it seems to have very limited use.
Bureaucrat: This is clearly for money & production while slowing down its opponents. Why it's not an attack, I have no idea. Definitely decent to get for the matching strategy.
Cellar: This definitely is useful early on to avoid those pesky Estates, and should have some use later in the game for money & action-based decks that can't or don't want to trash Coppers/Estates/Curses. Production doesn't seem to really care.
Chancellor: This is fool's gold. The only use this has is resetting your deck after you just recently used up valuable cards, and the +2 coin is just barely suboptimal compensation. I'd like it more if it was +3, even if it cost 4 to buy as a result.
Chapel: Anti-Witch, anyone? This is clearly around to keep your deck focused and minimize poor draws. Primary targets are obviously going to be cards that helped early on but do little in the late game, such as Copper and Militia.
Council Room: This one generates mixed feelings, as I suppose is the intent. No deck is going to want large amounts of this card, but it's hard to argue against its use in any deck.
Feast: I really hate this card, but that's probably because I don't work with a production-style deck. It's definitely an ideal card for such a deck to get off the ground initially, as all players are guaranteed the opportunity to get this within the first 2 turns.
Festival: I see no reason why any deck would reject this. It clearly offers a benefit to anyone that plays it. The only problem you get is that if you use a draw card and this comes up, it's worse than Silver. Honestly though, you can easily tell the style of play someone's going for as to whether they accept or refuse the first chance to grab it. This allows you to work around that knowledge.
Gardens: I'm not really sure what to make of this card. I understand it works for the + buy strategy, especially if the game is dragging on for a long time. If anything, it might merely be a cheaper Dutchy, which still gives it some value.
Laboratory: Given that this is what made me start writing, I feel this is remarkably good. This is great to have in a deck, but keep in mind the importance of tempo. A buy spent on this is a buy not spent on something else which may have a more direct effect on your play. This is clearly inferior to Mine in a money/production-based deck, for example.
Library: This looks like fool's gold to me. Clearly, only a deck focusing on action cards is going to benefit from this. By the same token, it needs to work in concert with a +2 action card preceding it to be most effective. Otherwise, it's almost less effective than Adventurer.
Market: This is for all those who just refuse to commit to anything, or perhaps want a deck with more balance than theme. The most useful part of this card is the extra buy, but it can still complement pretty much any deck. It's actually its lack of theme that makes it less than stellar.
Militia: Clearly, this is for the aggressive player right from the start. It doesn't have much use later in the game, but it gives early acceleration while decelerating everyone else (hopefully).
Mine: This is a key player in a production deck. Mine is absolutely how such a deck has to get going. Of course, a money-based deck can find some use in this if it finds itself at exactly 5 coin.
Moat: Primarily, you're holding this up as protection from other attack cards, because +2 cards doesn't get you very far. If the field is filled with attacks, this is definitely worth considering.
Moneylender: I see this going for a money deck more than production, but production can find some use with it early on.
Remodel: This can power a deck all by itself. It'll keep the deck focused while accelerating your capabilities. It takes some time to understand that trashing is not necessarily a negative thing, but if you've got the guts for it, this can reward you well.
Smithy: Good for money & action decks. Pathetic for production. Money decks like that they're probably getting more money out of it. Action decks use it either as their last play or in concert with +2 action cards.
Spy: Clearly this is best in action decks, and it is tailor-made for aggressive players. You use this to keep your opponents from getting off the ground too early.
Thief: Any deck can use this card based on the circumstances. Although it doesn't truly contribute much to an action deck, it still helps keep other people from escalating. It can easily make your deck into a money deck while you're at it.
Throne Room: You can really use it anywhere, but it's really at home in an action deck. After all, it needs to be with other actions to work at all, and a deck themed that way will use it most consistently.
Village: Definitely a centerpiece of an action deck, this is not necessarily something you pick up real early. You need other actions to make this work, so you grab it once you've gotten a few things already. Of course, like other cantrips, it does help keep your deck thin & focused.
Witch: If you wanna be aggressive, this is what you use. Cursing everybody will not only hurt their score, but also make their hands less consistent.
Woodcutter: This is definitely a fine early pick for a money-based deck, but only a money deck. Action decks don't need the extra buy, production just happens to buy because it can. Only money decks will realistically have the excess finances to use it.
Workshop: This is really bottom-of-the-barrel for production. However, it still adds stuff to the deck, which works wonders for Gardens.