Do NOT stick cloves in it and use it for besamim. Even though the idea sounds spiritually lofty, it is not permitted. One is not permitted to buy or sell (issur schorah) or change the appearance/shape (shinui tzurasah) of something that has kedushas shvi’is. Sticking cloves in an esrog is unquestionably changing its structure.
There is an opinion (the Ramban)that it is a mitzvah to eat peoros shvi’is including the esrog. Since (almost) nobody actually eats one, but many like to make jam out of it, this is permitted. One must make it hefker by the time that it is not readily available in the fields, which is around the month of Shvat.
If one leaves it lying around, it does not “rot”, it just dries up. This is because there is almost no fruit inside, and obviously is also why almost no one eats it, jam making being for the very industrious among us. When it is dried up, it no longer has kedusha as it is no longer a fruit. It may then be discarded in a respectful way, such as we do with all peiros shvi’is.
It is quite sad that in a highly Torah minded community such as YWN, so very little is known about this topic. Whether one lives in Israel or not, it is part of Torah and should be studied, more so than many of the other topics discussed here.