The “Begats” are sections of the Bible that go over family trees leading back to Adam and Eve. The word begat means “to have fathered” i.e. Noah begat Ham, Shem and Japeth. Moses is said to have written the first five books of the Bible, including a family tree from Adam to Noah.

“Begat deez nutz” is their way of announcing Santa – as more explicitly stated in the next line

This also refers to the Dr. Dre song “Deeez Nuuuts” on The Chronic (Snoop Dogg was a contributor to the album, including that song, and his appearance on The Chronic catapulted him to initial fame prior to his debut release of the critically acclaimed Doggystyle)

A play on the famous opening to Snoop’s “Gin and Juice”:

With so much drama in the L-B-C
It’s kinda hard being Snoop D-O-double-G

Israel BC refers to the time period (prior to 1 AD), and Moses of course was famous for directly conversing with God (while other prophets typically received one-way communication in the form of dreams). Also, LBC is Long Beach, California.

This might also refer to the Jewish practice of avoiding writing “God” directly (instead observant Jews will typically write “G-d”).
Snoop Dogg is known to spell out the D-O-double G

The sixth commandment:

Thou shalt not kill.
-Exodus 20:13

However:

And it came to pass in those days, when Moses was grown, that he went out unto his brethren, and looked on their burdens: and he spied an Egyptian smiting a Hebrew, one of his brethren. And he looked this way and that way, and when he saw that there was no man, he slew the Egyptian, and hid him in the sand.
-Exodus 2:11-12

Admittedly in Moses' defence, his murder of the Egyptian occurred well before he received the commandments from God.

This might also be a pun, given Moses literally broke the tablets:

And it came to pass, as soon as [Moses] came nigh unto the camp, that he saw the [golden] calf, and the dancing: and Moses' anger waxed hot, and he cast the tables out of his hands, and brake them beneath the mount.
-Exodus 32:19

Moses suggests that Santa is a pedophile, sneaking into houses with children to deliver presents, as well as monitoring who is naughty and nice. The use of “naughty” also has sexual undertones (and Moses' “stocking” suffice to say is a phallic euphemism)

This is also a reference to Snoop and Dr. Dre’s song “Nuthin' But a G Thang,” where Dre says:

Well I’m peeping and I’m creeping and I’m creep-in

These fake gangsters carry guns in their waistband with no intention to actually use them – Plaxico Burress style. They would rather be perceived as killers that can force people around them to obey their every command than actually kill people. In other words:

You say you a gangsta but you never pop nothin'
We say you a wanksta and you need to stop frontin'"

Dre is also referring to the Western movie genre trope of a gunman firing at the ground to make the victim “dance” (hop around trying to avoid being shot in the foot)

https://www.youtube.com/watch?v=yUe_Pi8NfT4

Robin is enthusiastic to help Batman, but Batman brushes him aside, and Robin is upset by this.

This references the recent trilogy of Batman films by Christopher Nolan. In the films, Batman has the habit of simply vanishing whenever he feels the conversation he is having with someone has concluded. He does this when people have their backs turned or aren’t looking, leading to frustration when they finally realize Batman is gone.

Watson is continuing the previous line by Holmes, implying that Holmes needs a doctor for a sidekick because Holmes' rap skills are ill. Ill is a slang term something that is good or of high quality.

Caroline asks if Big Boi is “balling” (if he is wealthy.) However, Big Boi disparages the gold-digging mentality as akin to prostitution.