Chocolate's comment and Kokoroatari's answer explains most things.
I think the etymology itself might explain something.
I would (appreciate it) if you could ...
れば sounds like a wish and can be used as a real wish rather than an indirect request. e.g. 直ぐにまたお会いできればと思います / せめてみんながこれを信じることができれば.
Could you ...
ませんか is an explicit request, just softer than imperatives
I wish you would ... (Could you ...)
～たいのですが is almost always followed by an explicit request, but it's omitted so that it sounds a little softer and you expect the listener will understand what you actually want to say.
I will rank the “directness” as: 頂ければと思います < 頂きたいんですが < 頂けませんか
けれども, けども, けれど and けど are much more causal than が. けども is more masculine, けれど is more feminine, けれども and けど are neuter. が is the most common choice when you are using keigo.