Avoiding war

You might have more experience on Crippling than me, but here are some answers anyway:

1) Does having a high diplomacy bonus keep the AI from being as likely to attack?

It helps in that it improves your relations with each AI. However it doesn't do that much, after all diplomacy is only one of many elements in the equation. Setting up 2 trade routes and getting economy / research alliances from an AI will also help a bit.

2) Can giving out gifts get the AI off your back?

In my experience it takes a lot of gifts to improve your relations with an AI. And all those gifts are only making the AI stronger.

3) Will giving in to their money demands keep them off your back for long? (I usually say "Screw you" when they demand cash from me)

It helps for a while; refusing to pay will only hasten their attacks. However even if you keep paying, at some point the AI will see you as a good piece of lunch and attack.

4) Or should I just try to ally up with a powerful civ?

Good move to build a good relationship with a powerful AI. And then the next logical step would be to bribe your powerful ally into attacking some other AIs (preferably before you're really allied, since allies will expect you to join in any war they are involved in). This reduces the chances of you being attacked, plus the ongoing war weakens all parties involved.

5) Or should I just never let my military fall behind the curve?

That is typically the best way to avoid war (barring any "the leader of civilization X visited your planet and was assassinated" events). Get to the spin control center as soon as possible if your civilization has access to it. Keep an eye on the AI military ratings and make sure you keep up - or pull ahead.

6) Or should I except the inevitable and get my ground forces as strong as possible?

Preferably not. Losing a few thousand inhabitants is quite expensive since it means less tax payers; it may be cheaper to build a bigger fleet and pay its maintenance costs.

If your ships aren't as good technology-wise, and your economy is stable, you could set funding to research and set all colonies to focus research. Depending on how many colonies you have, you could completely research one line of defense techs (shield, PD, or armor) in a few turns.

Once you got to where you want to be tech-wise, you can design a new ship. You then have a few options to get back ontop militarily:
1) Set production slider to 0% for some quick cash and upgrade on all your existing ships. Pay attention to how many weeks it'll take to upgrade them. If in a war, they might get destroyed before you finish upgrading.
2) Set production slider to 100% with 100% military focus, and build some new ships. Rush buy some if currently in a war. You can always lower the production slider down if need cash.

Playing with the production slider can stall your social production, so don't keep this up for too long. But it is helpful sometimes.

Also, if you research most of the diplomacy techs, and have more colonies than the AI, they sometimes will not attack. I think they're scared of your 'potential' or something. They become more tollerant of your influence. Haven't quite figured out the specifics. I've had games where I don't build any combat ships for well over 300 turns into the game on a huge gigantic map, and never had the AI attack me.

edit: Another thing that will help avoiding a war, is to trade some techs for the AI's Economic and Research Treaties. You don't want to give any of the AIs your treaties. Normally I'll trade all the 'Trade' techs, 'Soil Enhancement', and a few minor techs for one, and the life-support and sensor techs for the other. Because the AI gave you the treaty, it doesn't want the bad relations for the rest of the game it'd get if it declared war and broke the treaty. I think this alone, with all 9 major AIs and 8 minor AIs giving me their treaties, plays a major part in allowing me to go 300+ turns without a single military vessel.

edit2: Started a new game today, and seems I was playing on gigantic maps instead of huge.

Learning to keep the AI off your back when you have a weaker military is a key part of moving up the difficulty chain.  You can do it diplomatically by using their strenght against them.  Rather than giving them things as gifts - look for a strong civ and bribe it to declare war on a civ that slightly weaker, or a couple of weak civs. 

You want to civ you're bribing to get into a war that will result in a stalemate.  I frequently have anyone with a strong military engaged in at least two wars - then start building my military later in the game. 

To add to Noctilucus' responses:  (2) Gifts usually aren't effective for more than a couple of turns. (3) Paying Tribute isn't a bad idea if you can afford it - but it's sign that AI isn't busy enough and need to be engaged in a war with someone else. It's a sure sign you need to do something to avoid another request.  (5) You can be behind militarily - just keep checking the Report section on the Foreign relations tab to make sure the 'Too busy too go to war with you' condition is listed -- I've very rarely been attacked when that was listed.

Focusing on Trade and Diplomacy research can really help keep you out of an unwanted war.

Trading with the AI Civs will give you a good boost to your relations with them, and a sizeable bonus to your income. With 10 trade routes I can have my production slider at 100% and still rake in 1000+ bc every turn.

I then use all my excess money to trade for military/ship techs and bribe the AI to go to war. I'm currently playing a game in which I have done NO military research of my own at all. I've bought all my techs from the AI and I have the most powerful ships in the game I've seen.

Also, keep an eye on reasons for the relations you have with other races. If the most powerful races in a game are evil, then taking the evil morality will get them to like you more and be less likely to attack you without provocation. I've found a lot of success going with Nuetral morality early on so I get bonus to trade and nuetrality learning centers. The AI is more forgiving of a neutral races than a race of opposing morality.

By far, the best way to delay wars and also to take advantage of a superior diplomacy rating is to get the AIs all tied up in multiple wars amongst themselves.  There are actually multiple reasons for this, and you probably want those wars to begin before the end of the colony rush (second phase).  It slows their research some, they have to rush buy defenders,  it may open up previously-taken resource anomalies for you to grab, and it also (and I've not seen anyone comment on this next point) preserves the minor races for you to milk.

I generally compare the map and the positions of AI empires and try hard to get  and keep each one in at least two wars with races with whom they share space or border.