から is equivalent (about as much as you ever get between the two languages) to the English "from". It covers both the literal sense (a starting point for motion) and the metaphorical sense that's a rough synonym for "since" (i.e., the premises that lead to a conclusion).
Asking a question this way, while still polite, implies that making the request follows naturally from the circumstances.
んですが is (contracted) explanatory-の + ordinary polite copula です + sentence-connective が. Grammatically, the combination is roughly equivalent to ので (see also), but I would say the usage is fairly different. By phrasing things this way, you emphasize that the circumstances stand independently of the request.
In effect, this is more humble. English translations of sentences using sentence-connective が often turn into a single sentence using "although" or "but". That doesn't work here (since the circumstances aren't counter to the request), but it helps understand the nuance.
Let's take one example each.
時間がありませんから、急いでください。
"Since I don't have time, please hurry." There's a natural logic to it, and a presumption that the request is reasonable.
The other examples, similarly, request accommodation.
車を買いたいんですが、お金を貸してください。
"I'd like to buy a car, so please lend me (the) money (for it)." This is a considerable imposition, and apparently motivated by a want rather than a need.
The other examples, similarly, request a favour, perhaps to cover the speaker's prior error.