The fiery blade cannot be an improvised weapon
The rules for Improvised Weapons specify that they require an object:
An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.
The rules define an object as:
a discrete, inanimate item like a window, door, sword, book, table, chair, or stone
This is further elaborated on in this answer to the question "What is an object?".
While it can be argued that the fiery blade is discrete (I'd agree), and inanimate (a bit harder to convince of, but I'd agree as well), it is not an "item":
an individual article or unit
By the very nature of fire, you can't really refer to it as a unit.
In conclusion, the fiery blade you evoke from flame blade is a spell effect, not an object. Therefore it cannot be an improvised weapon.
What if my GM rules that it can be?
Allowing the flame blade to act like an improvised weapon opens up more mechanics than the spell accounts for and a GM will need to consider how that may impact the other aspects of the game. Considering this, if your GM does allow the spell effect to be used as an improvised weapon or decide that the fiery blade is an object, the following answers apply:
Would such an attack benefit from scimitar proficiency?
Maybe. The rules say:
At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
So it is up to the GM whether the fiery blade is similar enough. It has the same size and shape so that's a point in the "yes" direction, but it is made of fire rather than metal.
Would it deal 1d4, 1d6, or 3d6 damage?
It would deal either 1d4 or 1d6,
If your GM rules that it is similar to a scimitar, it will use the scimitar's damage. Otherwise it:
bears no resemblance to a weapon [and] deals 1d4 damage
So, in short it is up to your GM.
Would the damage be fire damage?
Maybe. As before, if it's being treated like a scimitar it has those properties:
can use a similar object as if it were that weapon
... and therefore it would deal slashing damage.
the DM assigns a damage type appropriate to the object
... and fire is probably the most likely candidate. Again, up to the GM.