The best thing for them is high calcium veggies. You can also use algae wafers or even regular fish flakes.
It varies by species, but a heater is generally not necessary. A heater just makes them grow faster and makes them somewhat more active. In cooler water, they don't move around as much, but either way they're fine.
A filter needs to be rated well for whatever size tank you have, just like with any other fish. The rule of thumb for crayfish is 1 per 10 gallons, unless you have dwarf crayfish (which from the pictures seems you might). But that rule is mainly for territorial/aggression purposes in larger crayfish. On that note, be aware crayfish are aggressive to fish and to one another... again unless you have dwarf crayfish.
Any good HOB filter will do the trick, but avoid undergravel filters as the crayfish might dig them up.
A bubbler is an absolute must for crayfish (or pretty much any livestock in an aquarium, imo).
The mother crayfish should have a nice place to hide while holding her eggs. It looks like you have some good rock caves in your tank, so make sure she's comfortable in her own cave. They won't typically eat while berried (holding eggs) either, so don't worry.
When they hatch, there may be some cannibalism, but if you want to save the babies, either provide a ton of hiding places for them in that tank or get another grow-out tank with adequate hiding spaces. And keep them well-fed.
