Having a beard is considered an honor (Shabbos 152a). There are many Shitos that hold having a beard is Halachicly obligatory. For a collection of advantages of having a beard from a Torah perspective, see Orchos Yoshor, by R. Chaim Kanievsky shlita, Ch. 5. There is also a Sefer called Hadras Panim Zokon that has a lot of information on this.
Trimming the beard is a Machlokes in the poskim. The Tzemach Tzadek and others prohibit even trimming it, but many others permit. There are Kabbalistic reasons for not trimming the beard at all.
Regarding Payos, there’s a machlokes if you can cut them, see Tiferes Yisroel (Makos 3:5) and R. Hillel Kalama (quoted in Shaul Sha’al 98) – who prohibit, and Chasam Sofer (OH 154) and others – who permit.
The Chasam Sofer (Haghos YD 181 quoted by his son Ksav Sofer) says that it is customary to let the payos grow long, down to the jaw. This is unnecessary, says the Chasam Sofer, but those who do it are considered holy.
The Arizal (quoted in Bais Lechem Yehuda YD 1818) says that the payos need not be longer than the bottom of the beard, and he would cut them when they reached there.
Maharasham, however says that he was told by R. Meir Promishlaner that he should never cut his payos, and it will be a segulah for Arichus Yomim.
Maharshal (Yam Shel Shlomo Yevamos 12:18) says it is wrothwhile not to cut the payos at all, since the exact measure for the payos is uncertain.
Mishna Brura (251:2, Biur Halachah) says that at least the hair from the temple until the bottom of the ear should not be cut, because it is a possible issur d’oraysa.
In any case, it is true that the “shiur” of the payos being able to fit neatly behind the ears and then being cut as they protrude from below the earlobe has no Halachic validity. It is just a style by certain segments of Klal Yisroel. They are using the Halachic shiur of the Mishna Brura (until the bottom of the ears), and the rest of it is for no real Halachic reason.
As far as putting the payos behind the ears, that began as a way to avoid anti-semitism in Europe from goyim who would harrass Jews with long payos. Nowadays at least in Eretz Yisroel there’s totally no reason for it, and in fact Rav Chaim Kanievsky shlita (Orchos Yoshor p.20) writes that it’s wrong, since it looks like youre embarrassed of the Mitvzah.