Should I always insert "and" between two verbs in imperative mode?

As far as I understand, the word and is usually inserted between two verbs used in imperative mood in English. For example, “Go and make me a drink.”

How obligatory is this? Can I claim that it is ungrammatical (or at least less typical) to have to consecutive verbs in imperative mood?

Also, what about the phrase “Go make me a drink”? Should I treat go and make in this phrase as two imperatives?

The verb go can take a bare infinitive following. So can come. (Edit: And sometimes, so can run.)

Examples: You can go get your hair cut or come get your supper, you can go see a movie or come see what I’ve got, you can go wash you car or come wash your hands, and you can go plow the field or come milk the cows with me. Or you can just plain go hang.

This is all perfectly normal. Here are some OED citations of this:

• 1813 Jane Austen Lett. (1884) II. 216 ― Your Streatham and my Bookham may go hang.
• 1831 S. Lover Paddy the Piper, Leg. & Stor. Irel. 151 ― There’s an iligant lock o’ straw, that you may go sleep in.
• 1912 Pedagogical Seminary XIX. 96 ― A rebuke to pride with the notion of ‘get out’,··‘Go jump in the lake.’
• 1922 Joyce Ulysses 750, ― I dont like a man you have to climb up to go get at.
• 1968 Globe & Mail (Toronto) 17 Feb. 35 (Advt.), ― Now there is polish as well as boyish personal appeal. Go see.
• 1968 Encounter Sept. 22/1 ― Go hit your head against the wall.
• 1969 A. Glyn Dragon Variation vii. 216 ― Let’s go get ourselves a drink.

And similarly with come:

• 1803 Scott ‘Bonnie Dundee’, ― Come fill up my cup, come fill up my can, Come saddle my horses and call out my men.

All those are different than going to see the doctor or coming to believe his story, because they don’t take to.

Edit

As I wrote in a comment, the verb run can sometimes do the same thing as come and go do.

Examples: Run tell your mother. Run find your father. Run get your gun. Run ring the dinner bell. Run close the gate. Run fetch the doctor. Run grab your hat. Run throw it over the edge.

When you have more than one verb stacked up like this, it’s the last one that counts. Any earlier ones just change the aspect. Contrast:

• Run go get your father.
• Go run this down to your father.
• Here, come run this down to your father for me.

In all those, it’s the last verb that is what is really happening, either getting or running. The pair run go seems to occur with some frequency, and takes yet another verb following it. Here’s one recent example:

“Alright then. I’m just gonna run go find my mom and let her know that I’ll be late getting in tonight,” Jennifer said before kissing Brant’s cheek. “I’m sure they’re together, so I’ll tell your mom too.” Jennifer smiled.

That’s am going to run go find, which is a huge whole lot of stacked verbs, but still it’s the last one the one that tells you what sort of main action is happening here.

            What good is sitting
            alone in your room
            Come hear the music play
            Life is a cabaret old chum
            Come to the cabaret

            Come taste the wine
            Come hear the band
            Come blow a horn
            Start celebrating right this way
            Your table’s waiting

English definitely requires an "and" between imperatives, except in certain cases. The phrase "Go [imperative]" is somewhat idiomatic, which is why "Go fuck yourself" or "Go have fun" are perfectly grammatical. However, no other verbs that behave this way are coming to mind. In most other situations, the 'and' is obligatory.

For example, "Sit and eat your vegetables" sounds much more natural than "Sit eat your vegetables."