Hi! In some cases, an if and only if proof will be a simple matter of running through some algebra for which all of the implications -conveniently- go both ways. Here, it would be absolutely fine to just preface any line of the working with $\iff$. You may wish to supplement some steps of working for which one of the implications might be non-obvious with a small note at the side to explain yourself.

Another type of question may not be that easy. The form of the proof may well be completely different depending on which way you are going. In this case it is wise to start by saying -for example- 'I assume the left hand side and so $\Rightarrow,...,\Rightarrow$ the right hand side'. And then doing the same for the other direction. This may involve algebraic or geometric justification, or just a well thought out argument in words.

In response to your second query, you are absolutely right in saying $X=Y\Rightarrow X^{2}=Y^{2}$ but the implication does not go the other way. In addition, the next statement you wrote down is correct: $(X=Y\, \text{or}\, X=-Y) \iff X^{2}=Y^{2}$. And yes; if your statements involve inverting functions (as your examples do here) then you do need that one-to-one correspondence, but obviously this only applies to these special cases.

Often, one way of an if and only if proof might be easier, and that might be why little detail is given in the mark scheme for some questions. In essence, lack of detail in a mark scheme can mean a more simple proof, or it might just be that there are many ways of approaching a question or part, at which point it becomes infeasible to write down all solution paths and it might just be left to an examiners discretion. In other words, there might be reasons other than the part is considered trivial.

A little side note - you may sometimes see the slightly confusing terms: 'necessary' and 'sufficient' conditions. Say we have a statement $P$. A necessary condition $Q$ is a statement such that $P\Rightarrow Q$, i.e. if $P$ is true then $Q$ must be true too as it is 'necessary' for $P$. A sufficient condition $R$ is a statement such that $R\Rightarrow P$, i.e. if $R$ is true then $P$ must be true too, as $R$ is 'sufficient' for $P$. You might be asked to find a 'necessary and sufficient condition $S$ for $P$. This involves much of what we have just discussed as it requires you to find a statement $S$ such that $P\iff S$.

Hope this helped. Cheers!