Hellooo
Im like completely new here (I just sighed up actually) but wanted to reply to this question...I was born and raised in Japan so fluent in Japanese.
As for ちょうだい, it's not a word only used by women and children. Sure, grown up male adults can use it, pretty much everyone can use it. But it's true that it might sounds a little childish, and you might not wanna use it in business scene.
お茶ちょうだい -- among kids, friends, family members. both male and female. more casual.
お茶ください ---- to people who are older than you, who you meet for the first time, who you are not that close to, etc. more formal.
As for ごらん、again, this one could be used by everyone.. pretty much.
Sure I can picture a granpa/grandma saying that to their grandchildren, but it could also be from mom to kids, friends to friends...but then again, you might wanna say してみてくださいto elderly people, just to be polite.
After all it simply means "Why don't you try 〜?" Like you introduce someone to the person for the first time...
And ..座ってごらん 座ってみて..
Hmm...My first response would be..."not much difference."
Both of them sounds like... "Come and sit here, something cool's gonna happen." Like you can expect something good is going to happen if you sit.
But personally I think I hear 〜してみてa little more often than 〜してごらん。
Hope this helped! :)
Rika