How would I translate "student of Klingon"?
Or "Klingon language student" may be closer to what the actual translation should be. The individual words are simple enough: ghojwI' and tlhIngan Hol, but how could these be strung together to mean "someone who is learning Klingon"
Bonus points: I've stated three different ways of expressing the same idea in English; how many ways might there be in Klingon?
nouns can simply be strung together with the same order as in English?
Basically, yes. There may be situations that are more complex, but most of the time, the modifying nouns just get placed before the main noun and a longer strung of nouns will wind up in the exact same order in both languages. This includes nouns with 's. In Klingon you don't add anything like the 's, but it still works the same:
the Klingon Admiral's ship's flight reaction control panel
tlhIngan 'aj Duj leng Dotlh SeHwI' pat
The word "of" in English reverses the order of the words in the same English phrase without "of", so watch out for that. "The metal sword of the emperor of Klingons" becomes tlhIngan voDleH baS yan "the Klingon emperor's metal sword".
Have a look at §3.4 "The noun–noun construction" in The Klingon Dictionary.
it is possible to combine nouns in the manner of a compound noun to produce a new construct even if it is not a legitimate compound noun ("legitimate" in the sense that it would be found in a dictionary).
The translation of two nouns combined in this way, say N1-N2 (that is, noun #1 followed by noun #2), would be N2 of the N1. For example, nuH weapon and pegh secret combine to form nuH pegh secret of the weapon. An alternate translation would be N1's N2, in this case, the weapon's secret. As discussed in section 3.3.4, this is the Klingon possessive construction for a noun possessed by another noun.
Basically, putting two nouns next to each other sometimes works very similarly in both Klingon and English, but it's not always 1:1.