Is it necessary to take full bath or shower after engaging in intercourse (or even a love bite and a smooch)?
No it's not allowed to pray right after intercourse unless you first took a bath.
Sex brings upon a state of being in ceremonial impurity and requires ghusl [Surah al-Maidah 5:6]
Once you have taken a bath, only then can you pray.
After intercourse it's good to have a bath if possible, if not only the washing can do too. If you had intercourse I presume you're at home so it's possible to have a bath.
After a kiss or love bite the bath is not obligatory, not even the washing.
Allah loves those who clean themselves a lot so it doesn't matter what you did, before praying I think washing is the best way to start any prayer or any Quran reading.