When do people who live in places where it is dark for days Daven Shachris and put on Tefillin?
Please provide a source (Mareh Makom).
See this excellent article by the Star-K's Rabbi Heber.
In summary, if you're close to the North (or South) Pole and it's dark for days, the opinions are:
If you're ever stuck in a situation like this for real, ask your rabbi which of these opinions (or combination thereof) to follow! See the article for Rabbi Heinemann's psak combining some of these.
With regards to space travel, I recall hearing another opinion that you revert to Jerusalem time, but I don't recall the source.
I recollect that friend of mine told me that his father-in-law was stationed in Greenland and was told by R' Moshe (or by someone who asked for him?) that he should go by the closest city with normal times.
I haven't done a whole lot of research on this subject, but off the cuff, perhaps all of these times aren't "real times", but a personal gauge. According to the Maharsha 26b, even Rabbi Yose b' R' Chanina, who holds tefilos keneged avos tiknum, also held of k'neged korbanos tiknun, but Chazal would not have established the tefilos at those korbanos times if not for the original tefilos established by the Avos who fulfilled the pasuk of erev vavoker v'tzaharayim asicha... So it could be that absent the times of chazal, the halacha would revert to 3x/day even without hard zemanim. Although the pasuk says specifically evening, morning and afternoon, it's possible to say that's lav davka as Tosfos proves that Yaakov davened maariv during the day (for Tosfos a proof that the halacha is like R' Yehuda). Lulei mistafina midivrei Tosfos, I would have said the timing mentioned in the pasuk is lav davka (and this is how the rabanan who argue w/ R"Y would answer Tosfos' question).
Krias Shema is a little fuzzier. Are the zemanim of 3 hours, etc. mideoraysa, or applications of Uvshochb'cha uvkumecha. In other words, can we say in absence of zemanim, we revert to shechiva and kima of your time zone? It seems acc. to the opinions above that that wouldn't be the case.