With chickens it's usually not a case of "need to" but more of a "can if you wish". There are exceptions of course, but not with the stuff yo are asking. There are all kinds of different ways to do the things you are talking about.
So I just got 4 baby chicks, about 3 weeks old. Do I need to handle them?
What are your goals? If you want them to be pets you probably should start now, though others achieve that with chicks a lot older. The main ingredients for that is food and patience, but a calm demeanor helps too. If you are OK with them not climbing into your lap or riding on your shoulders it still helps to get them used to you but it's not as critical.
Can I feed them treats from the kitchen yet?
A broody hen feeds her chicks all kinds of treats shortly after she takes them off the nest so of course you can. But usually before that she takes them somewhere that they can eat some grit. That's usually a bare spot where they can peck at the ground. You probably already know this but they use small rocks in their gizzards to grind up food that needs to be ground. If you are in the US we call this grit. In the UK or some other places this is called insoluble grit. Soluble grit is oyster shell and you do not want to use that for chicks. If you modify your profile to show your general location that can help us with several questions.
Do I need to let them in the backyard or can they stay in the bin until I put them in the coop?
An excellent plan! This gives them access to grit and helps them get acclimated to the weather. They feather out faster and may get started on certain immunities in your environment they might need later. Again it would help some to know your weather but I've seen a broody hen take her chicks out to eat in temperatures below freezing. They run around a lot more than people would expect and them go back to Mama to warm up. You need to be there to monitor them and Rose's idea of a pen to make them easy to capture is a great idea.
Can rabbits and chicken share a coop?
I don't have any experience with this but some people on here do. Their needs are going to be a little different so it may depend on how the coop is set up. Sometimes they get along and sometimes they don't. I've seen some set-ups where they share a run but have separate coops. It may be a trial and error thing to see what works for you. I think there is a rabbit section to this forum, you might try asking in there.
On another note, what's the best way to introduce my dog to these little guys
My brooder is in the coop. The way we trained the dogs was to regularly take them on a leash to the coop and let them sniff the chicks through the brooder wire sides. We did this as lot. Then when the chicks were let outside I made sure I was outside when the dogs and chickens initially met, they shared territory. The dogs had been trained that "no" meant no. I was kind of verbally abusive to the dogs a few times, especially one of them. One time I smacked that one across the rump with my hat to emphasize "no". Your biggest risk is usually when the chickens run from the dogs, that can trigger a prey drive. I managed to get them to the point where they could share territory but it took effort.
Huskies are known to have a strong prey driven so yours may be a bit challenging. But if you can get him to accept them as part of his pack they might get along well or maybe he will just ignore them. So start introducing them as soon as you can but under controlled conditions. Don't trust him for quite a while.
