I've known the Stones song for at least 40 years. At least, I thought I knew it . . . until I heard this . . . this JEWEL! I am so glad I stumbled on this. Thank you!

So. There are a few issues with usage of words in everyday English. Let me put some work into it.

Anger Mary > This isn't quite natural English. I would go with Angry Mary.

so pay good attention > so pay close attention

I've already told it to you > I've already told you

a lot, lot of patience > a lot, a lot of patience

will see to join the dance > will see joining the dance
"coming to the dance" would also work.

lightning in the eyes > lightning in her eyes

I know which one will be more afraid of the two us > I know which of the two of us will be more afraid

too gentle to you > too gentle with you

If you insist on looking at the other girls > If you keep on looking at other girls

with all claws lowered > with all claws unsheathed
I suppose you could just say "all claws out" as well, but "unsheathed" is the word that applies to the situation.

But do not rely on it too much, a good advice, take care > But do not rely on that too much, heed well this good advice
Or you could say, "take this advice to heart."

Anger Mary is now there, > Angry Mary is now here,

Soft Mary is already not more than a memory > Soft Mary is already little more than a memory
Or " . . . already nothing more than . . . "