Throw In Reverse

is he sleep

"He threw the car in reverse."
"He put the car in reverse."

Does "threw" imply haste?

For the most, part I would say yes.
However, I am certain that there are some contexts in which it could be used and the idea of "haste" might not apply.

"He threw in reverse and backed out of the parking lot."

This was ruled non-standard English, because "threw" is correctly used intransitively. But in street talk, does it explicitly suggest haste (and style)?

In "street talk", I'd say yes.

So,

"He threw in reverse."
is non-standard English, and,

"He threw the car in reverse."
is standard English?

He threw the car in reverse.
He threw it in reverse.
He threw the gears in reverse.
All of the above are correct.
You need to tell the reader "what" he threw into reverse.

Would:

"The car threw in reverse."
be correct, in the sense of:

"The car went in reverse."
?

"The car threw in reverse" does not mean "The car went in reverse."
A car cannot throw anything.

Throw pertains to changing gears in the automobile's transmission.

The driver of the vehicle changes the gears into one of the forward gears or the reverse gear. If the automobile has an automatic transmission, the driver can put the car in "drive" or "reverse" or "park".