1a) Yes, it's correct.
1b) "..., ob das Wetter ... besser sein wird" is also correct. "Sind" is wrong, it's "werden + Infinitiv". One version uses "sein" (to be) as main verb, the other "werden" (to become). "Werden" can be used both as a main verb and as an auxiliary verb for the future tense.
2) No, you can't rewrite it that way. Dependent infinitives are always placed at the end, in reverse order. In a main clause, you can put the conjugated verb ("möchte") in second position, but never an infinitive. And "ob" indicates that it's a subclause, not a main clause, so there are two reasons why you can't do it. :-)
3a) "Sie wohnt (!) aus diesem Grund bei einer Familie."
Main clause, conjugated verb in second position. "She" = singular. "They" = plural.
3b) "Aus diesem Grund wohnt (!) sie (!) bei einer Familie."
Main clause, conjugated verb in second position, "aus diesem Grund" in first position, so the subject moves after the verb. English needs SVO word order, so English uses a comma: "For this reason, they live ...". German doesn't work this way, you can move most parts of the sentence around (except the verb). That means there is NEVER a comma in such cases.
3c) "Sie wohnt (!) bei einer Familie aus diesem Grund." is very unusual. As Eliot has said, the order is time - manner - place (in English, it's place - time, so be careful). So, under normal circumstances one wouldn't use it.