Ich habe morgens einen Deutschentest genommen.
In German, you don't take tests, you have them. Also, "morgens" tends to mean "every morning". If you want to express "in the morning" with reference to this morning, use "am Morgen". (This may be a feature of my idiolect, but maybe some others will confirm...)
Obwohl ich [gern Kuchen / Kuchen gern] esse, bin ich allergisch dagegen.
Which of these two is considered (more) correct?
Go with the first. The second happens to be possible in this context, but it's generally much more restricted in its use.
Obwohl ich [gern, Kuchen, morgens, in meiner Küchen] esse, bin ich allergisch dagegen.
I'm trying to say "Although I like to eat cake in the morning in my kitchen, I am allergic to it." A very situational sentence, I know, but I'm trying to figure out word order
Here things get extremely tricky, because word order in German is heavily dependent on meaning and the discourse situation. One correct order for your sentence would be "obwohl ich gern morgens ins meiner Küche esse". "morgens gern in meiner Küche kuchen" is also possible, though. The difference here is really semantic, but it happens to be minimal. In the first version, what is like to do is eating cake in your kitchen in the morning. In the second, you that it's the case in the morning that you like eating cake in your kitchen.
The basic order is time > place > manner > DO. However, it can be modified. First according to considerations like the one above. (The difference there is called a difference in "scope" of the adverb "gern".) Second, and more complicatedly, it depends on whether the DO is indefinite or definite, and whether it has been mentioned in the discourse or not. I'll give you the general rules here, although I'm afraid they're not the full picture. I'm not including intonation, for instance, which also plays a role. Maybe later...
Rule for definites (i.e. pronouns, proper names, noun phrases with the definite article, with deictic pronouns...):
If it's familiar (i.e. has been mentioned), put it to the left of all adverbs. As an example, take the following exchange (I'm using a full noun phrase instead of a pronoun, which would of course be much more natural, just for illustration):
A: Was ist eigentlich mit dem Peter?
B: Ich hab' den Peter gestern im Park getroffen!
NOT: Ich hab' gestern im Park den Peter getroffen!
If it's not familiar, put it to the right:
A: Stell' dir vor was passiert ist! Ich hab' gestern im Park den Peter getroffen.
Rules for indefinites:
You will generally keep those to the right, because usually they will introduce new entities in the discourse, i.e. be unfamiliar. However, with partitive existentials, you can sometimes have them to the left:
A: Was ist eigentlich mit den Schranz-Brüdern?
B: Ich hab' einen von denen gestern im Park getroffen!
Weil ich dieses Sommer in Europa reise, werde ich ein reisender Student [sein]. (side question, is sein optional here?)
The sentence is grammatically correct. "sein" is not optional here, though. It's only optional with "werden" when you are talking about a property that the subject will have for a long time, that will be a defining characteristic of him, etc. So if you're talking about a child growing up and you want to say that she'll become a tall girl, then you can say "sie wird groß". Or when you'll talking about a cake that you're baking, you can say "das wird gut". If you say that you will "reisender Student werden", then what you imply is that "reisender Student" is a profession that you intend to take.
Reisend ist wichtig (alternatively, would I say "Das Reisen ist wichtig"?)
Neither actually. The first is simply ungrammatical, the second is unusual. You'd just say "Reisen ist wichtig", or, if you want to be more verbose, "Es ist wichtig, zu reisen". (The difference is just the same as between "Travelling is important" and "It's important to travel".)
Ich kenne ein Mann, der ein guter König ist.
No, but only because the accusative is "einen
Ich kenne einen Mann, welcher ein guter König ist
That's correct, but it sounds weird. "welche" as a relative pronoun is very archaic.
In what cases would I use mögen or gefällen? I have been told that mögen is all around general liking, while gefällen expresses aesthetic liking. I'm not sure.
That's true. "gefallen" you can use for a piece of music, a theatre play, a movie (with those it's more natural than mögen). When you use it for a person, you're most likely (depending on the precise context and also the area of the German-speaking world that you're in) not expressing that you like them, but that you find them attractive. You also don't use "gefallen" for food, except to convey that you find the visual arrangement pleasing.