Great question! I must admit that I've never thought about it before, but now I've had a bit of a think (and done some sleuthing on the interwebs), and this is what I've come up with. You are correct in your prediction regarding words for east and west. All those words exist. This is where the tricky part begins... "Nord", "syd", "öst" (or "ost", a loanword from Low German), and "väst" are used regarding an exact cardinal direction, eg. “nord”, "nordöst", "sydväst", etc., or in compound words like "Nordeuropa" and "Sydamerika" that represent a single "entity". "Norr", "söder", "öster", and "väster" describe relative (outside) positions if combined with "om", as in your example "norr om Europa" (north and outside of Europe). Also, in answering the questions "where to" or "where from", these forms can be suffixed with "-ut" and "-ifrån" respectively, eg. "söderut", towards the south, and "österifrån", from the east. Note that this only works for "pure" directions. When talking about the southwest, for example, it would be "sydväst om", "mot sydväst", and "från sydväst". "Norra", "södra", "östra", and "västra" serve two purposes, as I see it. The first is to give the relative position within a geographic area, eg. "norra Sverige", "södra USA", etc. This usage is basically analogous to the English "-ern" suffix. Additionally, this is the form you will find whenever a noun is in the definite, such as "Norra ishavet", "Södra Vägen", etc. There is one archaic form that you might run in to as well, that I thought I might mention. These are "nordan", "sunnan", "östan", and "västan". These are old accusative forms with the meaning "from the <direction>". You only see these nowadays in fossilized expressions such as "sunnanvind" for a wind blowing from the south. This is by no means meant to be the exhaustive, end-all answer to your question, only my musings on the subject. Any other Swedish speakers are encouraged to share their thoughts (and corrections) as well. :)

Just thought I'd mention that two of your examples might not mean what you think they mean... "Vi ska åka till norr om Europa" means that you are traveling to north of Europe, i.e. you will not be in Europe, but instead be going to a destination outside (and north of) Europe. "Texas ligger söder om Amerika" means that Texas is south of America, and not in it. If you mean that it is in the southern United States, it should instead read "Texas ligger i södra Amerika" or, better still, "Texas ligger i södra USA".