Free Wi-Fi / available power / cashless food facilities in LAX airport?

Speaking from personal experience:

Firstly, I assume I'll be spending some of that time in customs, getting my bag, being searched, probed and whatnot.

If you are a US citizen, you'll probably spend about an hour with immigration and customs. If you're not, wait times could be up to four hours during a busy period, and that assumes you're not suspicious. Theoretically, this has gotten a bit better in recent years, but do expect to spend time waiting in line for them to process everyone.

Is there free wifi anywhere in the airport?

Not really, unless you have or pay for access to an airline lounge. Even then, you should check with the lounge staff as only a few have free Wi-Fi.

Are there power recharge points?

Yes. Every terminal is equipped with easily accessible power outlets.

Airside (the secure area), there are power stations for charging mobile phones and other small electronics in the domestic terminals, and there are sockets in the ground in every terminal.

Landside (non-secure area), there are usually a few in the walls but they're not exactly near chairs and the like. I'm also not sure that the ones landside are intended for public use, but the worst case scenario is that a security officer asks you to leave, so it's probably worth a shot if you're desperate.

Is there any food where I don't need American cash to pay for it (ie can use card)?

The vast majority of shops will accept credit or debit cards (Visa, Mastercard, AMEX, Discover are the common ones, but some will only take Visa and Mastercard). There's also a currency exchange in the international terminal, if you need one.

I can't answer for the free wifi or power charge points. Usually for electrical plugs, you'll see people sitting on the floor near a wall. All food facilities will take a credit card as long as it's VISA or MasterCard. Discover and American Express are less widely accepted.

Update: Here's a link to LAX on wifi access: LAX - Airport Information - Internet Services

In general wifi is not free except in some first/business class lounges.

I haven't been able to find any information on electrical outlets.