Yes: To "bail out" is to abandon a crashing aircraft by jumping out and using your parachute to descend to the ground. Hence the colloquialism "to bail" is to leave a situation urgently. So in the joke, Batman is "bailing" from church. We presume the church is a Christian one.
The other meaning that makes it a pun is that the actor Christian Bale played Batman in The Dark Knight (https://en.wikipedia.org/wiki/Christian_Bale).
"Nuts" is slang for "testicles". He's corrupting everyone's speech to say he's putting his testicles on their chin. It's immature humor.
nuts
noun [ plural ]
offensive for testicle
(Cambridge Dictionary)
Notice that the colleague (Key) is serious and concerned about the behavior of Vince (Peele), but Vince responds in an unexpected, immature way:
Colleague: Vince, I'm worried about you.
Vince: ... I was worried about these nuts on your chin.
Colleague: Vince, seriously.
Vince: These nuts were seriously on your chin.
"These nuts", often spoken as "deez nuts", was a fairly popular punchline for immature jokes a few years ago. There's even an entry on knowyourmeme.com:
About
Deez Nuts is the punchline of a conversational joke that involves asking someone a vaguely-phrased question to solicit a follow-up question in response, typically in the form of one of the five Ws, before yelling out the said phrase in an obnoxious manner. While originally introduced as a skit track on Dr. Dre's 1992 rap album Chronic, the joke saw a huge resurgence after it was featured in a short comedic sketch by Internet comedian WelvenDaGreat in early 2015.
It's not as widely used these days, but I imagine you'll still hear it in schoolyard jokes.
I don't know if "these nuts on your chin" (the whole phrase) was the comedians' own creation, or a well-known punchline like "deez nuts", but I can imagine that the idea would be universally understood for immature comedic purposes, or humiliation. Indeed, a contemporary example is tea-bagging in video games (squatting your character over a defeated opponent's face).
Best Answer
Its an old joke. The "old boys" are bragging about the size of their penises
"The water's cold" implies "my penis is long enough to water in the river below".
"It's deep too" implies "my penis is long enough to reach the bottom of the river."
If this is from the film I think it's from, you're going to have a hard time if you try to understand everything. There is a lot of Arkansas "good ol' boy" dialect. You'll enjoy it better if you just accept that you'll only get the gist. In this case the gist is "It is a dirty joke". But if you really want an analysis: Someone has actually written a thesis: Gendered sense of humor as expressed through aesthetic typifications