Grammar – Difference Between ‘I Don’t Remember the Name’ vs ‘I Can’t Remember the Name’

I would say neither of these sentences is correct English grammar. The first one might be correct if the two sons somehow became a single doctor. The second one is just incoherent. Also, you need "became", since this is in the past.

I think you're trying to draw a distinction between:

1. He has two sons who both became doctors.

2. He has two sons; both became doctors.

In this case, the first one leaves open the possibility that he has more sons. The second does not, since it makes "He has two sons" it's own thought. Strictly speaking, you could say "He has two sons" to mean he has at least two sons, but nobody does that.

(Like the old puzzle, "A man has two ordinary American coins that total 35 cents. One of them is not a quarter. What are they?" The gag answer: "The one that's not a quarter is a dime, the other one is a quarter.")

You speak no English (= “You don’t speak any English”) means, literally, “You have no ability to speak English” rather than “You don’t speak any kind or dialect of English”.

In the context of the song, however, You speak no English is a metaphor (or, arguably, a synechdoche) for “You are ignorant and inarticulate”—specifically, “You have no artistic sensibility”.

Four niggas in a whip
All foes say they ain’t fucking with Trice shit
Somebody’s lying, I’m undenying, denying me means that
The driver got influence over you on certain things
Be distinguished, just cuz you ain’t feeling his favorite emcee
Don’t mean you speak no English
Hop out the car, catch the bus
At least you be established as the man that you are

Obie Trice invites his listeners to assert their own musical preferences and renounce the ‘influence’ of those who have achieved material success (a whip is a Mercedes or similar expensive, high-status car) when they sneer at Obie’s music.

Just because you’re unimpressed by their favorite artists doesn’t mean that you have no musical taste.