First Question: The age they can go outside depends on the weather - temperatures. It also depends on whether you have a place set up that they can not squeeze out of and get lost and in harms way. - or will not be near mean older hens who would peck at them. They can go outside at 1 week of age if it is over 85 and you have them in a small enclosure and bring them back in if it drops really cold. They can stay outside at any age if you have enough baby chicks and a heat lamp and a set up that keeps them comfortable and safe. Your second question: I have never tried it but I have been told if you have an older hen who is broody and has been consistently broody ( where she peeks at you if you try to get her off the nest) then wait til night, slide a few one day old chicks under her- if she coos and takes them under her wings she has accepted them. If the babies sound distressed or the mother hen does not settle down- if she makes any noise other than a soft coo- or moves a lot and does not settle down the babies need to be immediately taken away. and even if all goes well you may want to hang around a while and go back and check in the night and be there first thing morning light to see if they are still okay. one thing this website does is offer so much more information under the table " Articles." click on those links and explore - there is so much information here that will help you make these decisions. Good luck.
Those are two different things.
The criteria for age at which a chick can go outside is what is the ambient temperature highs and lows outside and how many chicks can cuddle together.
You can't stick a chick under any random hen. It would have to be a truly committed broody hen and she would need to be close to the 3 week period of broodiness and as a general rule, chicks would need to be under 36 hours since hatch. Longer than that, the chick won't likely imprint on the hen as they gain independence.
Outside, in a safe place, with a heat lamp: from the first day. (Brooding outside)
Outside, in a safe place, no heat lamp: depends on temperature.
--once they're a few days old, you can take them outside to play briefly. Watch, and take them in when they sound distressed or start to huddle together.
--once they have all their feathers (around 3-5 weeks), they should be fine outside anytime it's above about 50 degrees fahrenheit. After they've been outside a few nights, they'll be fine even if it gets colder than that.
--if outside is as warm as the recommended brooder temperature for the chicks' age, they can be outside for any length of time, even before they get all their feathers (95 degrees the first week, 90 the second week, etc)