What can the writer assume in a proof? When writing a proof, what level of mathematical understanding can I assume my reader has?
For example, can I assume they know all odd integers can be represented by $2q+1$? (Right?)
Or that all even integers can be represented by $2k$?
When do I have the power to draw on a theorem?
Can I always assume that my previous problem was right and cite it to help me with my current proof?
 A: In mathematical writing, as in all writing, you need to think carefully about who your readers will be (and who you want them to be).  It is rarely possible to write in a way which will be equally satisfactory to every conceivable person.
In this case, although you don't say so explicitly, it sounds like you are writing up problem sets for a course, so your intended reader is the grader of the problem set and/or the course instructor (perhaps the same person).  With such a small audience it is feasible to simply ask them about their preferences, and while you probably don't want to do this before you turn in every single problem set, some questions in the beginning will probably make things go smoothly.  In general though, this type of reader is someone who knows the material well -- probably better than you do -- and in most cases broadly knows what you are trying to say even before you say it.  However, they are also looking for gaps and mistakes in your arguments.  So in this case I would start by erring on the side of including more details / supporting reasoning, while understanding that if you do not express any given idea / argument in the best possible way you are more likely to be understood anyway than with a reader who really doesn't know what you're trying to tell her.
With regard to your specific questions:

can I assume they know all odd integers can be represented by $4q+1$[?]

Gosh, I hope not, since this is not true: e.g. $3 \neq 4q + 1$.
[Added: It seems that the "$4$" was just a typo which has since been corrected to $2$.  In this case there may well be something to prove.  It is relatively common to define an integer to be odd if it isn't even, and then one has to justify that an odd number is of the form $2k+1$.  In fact, in an honors course for future math  majors that I am currently teaching this came up in the first week. I defined an integer to be even if it is of the form $2k$ for some integer $k$ and odd if it is of the form $2k+1$, but there was still something to show: every integer is either even or odd and not both.  The "not both" is easy, but the first part requires something: a few days later I proved it by mathematical induction, and then later stated it as a special case of the theorem about division with remainder. So no, for my intended audience I did not want to just assume that familiar facts about even and odd numbers are true, although I probably would do so in a course pitched either at a higher or a lower level.]

Or that all even integers can be represented by $2k$?

This is a standard definition of an even number (in fact, I can't think of any other standard definition at the moment). So this may well have come up before in class or in the course text.  If not, and you are not working with any other definition of even, then you can just say something like "if $x$ is even -- that is, $x$ is of the form $2k$ for some integer $k$ -- ..." and move on.  

Can I always assume that my previous problem was right and cite it to help me with my current proof?

I don't see why not.  But if you are unsure, ask your grader/instructor.
