I can try to help you with the first one, having just turned in the papers myself.
The first step is to get your papers together. That would mean, your transcript/s, your diploma/s, and the dreaded Apostille of the Hague for all of the above, individually. You need to have original copies, which for me meant sending a notarized letter (got that done at the U.S. embassy-30 euros for one, 50 for 2 documents), which I then faxed to the U.S., requesting to have official transcripts, as well as original diplomas released to a family friend, who got the apostille done for me, then fedexed it all to my address here (perhaps you have all of these things already here with you, but my process seems to have been the most complicated ever!). Then you get all of it translated by an official translator, which can cost quite a bit, but is something you would have to do anyway if you were wanting to apply to a public university. Once you have all of these, make photocopies of EVERYTHING, and carry the originals with you. Make sure you fill out the form which you can find online for the homologation (clearly I´m making up words now) of foreign degrees (you can find this on the ministry of education website). Make sure you obtain tasa 790, code 92, which you can pay at your bank, they will take a copy, and give you the other 2 copies. So these things you will then turn in to the office which, if you are in Madrid, is located at Los Madrazo 17, directly behind the Alcalá 36 building (which incidentally is where they can give you the tasa form-I went all around town looking for it, only to be handed it in person). The person present will take a look at all of your originals and photocopies, make sure you have everything, and take the photocopies of everything, a copy of the tasa, and a copy of the online form. I hope that made sense, and good luck-I am told this process can last a couple of years, although the lady working at the office told me to expect to get word from them in 3-4 months.
By the way, in order to work in the public sector, you need to have all of this done-private schools don´t have that mandate, to my knowledge.
Not quite sure about seeking word and residency permits, but I think your best bet would be to get empadronado first.
Good luck! Keep us posted if you learn anything new concerning any of your 3 points!
