When using any cryptosystem that relies on the Decisional Diffie-Hellman assumption (e.g., ElGamal, ECIES, Diffie-Hellman key exchange, etc.) then you need a group of prime order. Note that $\mathbb Z_p^*$ where $p$ is prime has order $p-1$. However, if $p=rq+1$ where $q$ is also a large prime, then you have a subgroup of $\mathbb Z_p^*$ of order $q$. If you take $r=2$, then these are the so-called safe primes (once it was thought that for RSA it's best to use these types of primes, and that's why they are called "safe"; however, today, it is accepted that it's best to just choose random primes for RSA). In order to get a subgroup of order $q$, indeed you just take the squares.
Answering your specific questions:
- For the cryptosystems that I mentioned above, YES you MUST work in the subgroup.
- You need to map any value that you need into the subgroup. You can map any value between 1 and q-1 by just squaring it.
- No, you do not do $\bmod q$ on the elements (you do $\bmod q$ when working on values in the exponent).
I recommend that you take a book that gives the number-theory background necessary for crypto...