TITLE: Generic elementary group theory problems.
QUESTION [23 upvotes]: This question is about generic group theory problems. here are examples for what I’m referring to:

Prove that any group of order $p^2$, where $p$ is a prime, is
  abelian.



Let $G$ be a group of order $2n$. Suppose that half of the elements
  of $G$ are order $2$, and the other half form a subgroup $H$ of order
  $n$. Prove that $H$ is of odd order ($n$ is odd) and that it's
  abelian.



"Let $G$ be a group with $|G|=pq$ for some $p$, $q$ primes such that
  $p>q,q∤p−1$. Prove $G$ is cyclic."


Most of the time I find myself attacking these kind of problems with no coherent strategy, just throwing all my ammunition (Cauchy theorem, Lagrange theorem, index 2 theorem, intersection unions and multiplication of (normal-) subgroups etc.).
Best case scenario is I manage to prove the statement yet I don't quite get why the statement is true because the proof is so long and involves a lot of cases and assumptions by contradiction that i can't see the forest for the trees.
A lot of the times there are more than one way to prove the statement which are not so similar. 
For an excellent example to what I’m referring to, look here.
Since I feel that most of the time I’m just juggling variables I'd like to know what should i do to understand what's really going on?
Another different thing that might ease my mind is an exhaustive list of the theorems that can be used to solve these kind of problems. That way i will at least know what are all the possible techniques that might work for these problems.
ADDED: Although neat proofs for the specific problems i posted are welcome they are not the reason i asked this question. I solved these problems and others too yet my proofs were long and lacked an identifiable idea. What I'm looking for is a general principle that could guide me in constructing proofs for these problems.

REPLY [2 votes]: First can I offer some encouragement: you said "I manage to prove the statement yet I don't quite get why the statement is true", well don't worry - we all feel like that a lot of the time!!!  If you keep going in the subject and then come back to the more elementary results later you will often find you say "OHHH I get it now".  This is much the same as what @Giorgio said.
One more practical suggestion I would like to make: when you have finished a proof, in some sense it's a huge psychological relief.  You've been working at the problem for hours (days... months... years... a whole lifetime maybe) and you are so glad to have finished it that you just want to go on to the next thing.  BUT if you stop, and read through the proof really critically - imagine you are trying to understand someone else's proof - you will often find that you are able to edit it.  Maybe just make the explanations a bit clearer, maybe find a short cut, maybe realise that two cases are actually the same; perhaps this will start you thinking of a whole different approach.  Slowly and bit by bit you can gradually get the proof into the best possible form.  AND you will probably find your intuitive grasp of the problem increasing.