I'm also fairly sure the different non-magic-dart arrow spells have the same costs (except chaos).
Now that I am back by the computer i could check and at low level its 8 for nether and 10 for the others. chaos is slightly harder to cast but that shouldn't matter as one gets 100% soon anyway.
I think you can only summon with more than 4k faith so if you just prayed for the final gift, you have to sacrifice a bit more to get past 4k again. Not sure, consult wikia discussion if needed.
Yes, one needs the same faith as for the god weapon to summon, But I couldn't find if it costs faith to summon or not, which was what I asked. And do they stay around only until one exit the area, until November is over or until next November, which is when one can summon another god.