Is writing short proofs inline bad form?
Maybe this is "opinion based", but here is what I prefer, and I think it is pretty standard.

I would only use the QED box  at the end of a proof environment (which started with Proof) which is preceded by a numbered Theorem / Proposition  / Lemma / Corollary etc.  That way, there is a clear statement of what is being proved, and the statement is clearly distinguished from the proof.  Subconsciously, I think of Proof and  as matched delimiters.

I don't think "too simple" is the main consideration for whether to use Proposition or Remark.  Rather, I think Remark should be used only for statements or discussions which are a digression from the main argument of the paper.  If you have a statement which is important to the paper, even if the proof is very simple, make it a Proposition or Lemma or Corollary (with a number), and follow it with a Proof.  (You might be able to get away with saying something like "we have the following proposition whose proof is obvious", and then omitting the Proof environment, but this can be annoying.)

In a Remark, if discussing a particular claim, you might include a short proof or sketch or outline.  Generally this would be written into the prose of the paragraph and not set off as a separate proof, and I would not use the QED symbol here.


  Remark. The previous result raises the question of whether every snark is a boojum.  In fact, the answer is yes, and this can be seen by noting that cheese is green and the sky is mauve.  However, it is still not known whether every boojum is a snark.


This would be an alternative to:


  Proposition 6.37. Every snark is a boojum.
  
  Proof. Cheese is green and the sky is mauve.