Following my accident, does my left shifter need to be replaced as the area could crack?
Based on your description and photo, probably not. As long as it still functions correctly. That is the key.
So, if it still works and is not broken/cracked, go ahead and use it. Keep an eye on it to see if a crack forms or the performance degrades, but lacking those things, what you have is a blemish.
Other things to consider: Are there any sharp edges that have been created post-crash that could cut your fingers/hand? If so, can they be resolved with a light filing or sandpaper/emory cloth?
No - that looks like cosmetic damage.
If it bugs you, then the small plastic faceplate is available new as a replacement part.
The roughness of the aluminium casing can be filed flatter, and you can apply some enamel paint to hide the file marks.
Likewise you could simply scuff and paint the existing plastic cover - it could be done to the same shade as your bike, or in a complimentary colour
Fortunately the rubber hoods look undamaged - they tend to be harder to fix.
Lastly - do a close once-over of the rest of the bike looking for other damage. Anything that pokes out is a candidate, so pedals and saddle etc.
That’s normal and harmless. Even much deeper abrasions in that location are just cosmetic.
If you’ve clamped the brifter with the correct torque (i.e. relatively weakly) it should twist around the handlebar if sideways impact forces get too strong.
I’d check around the pivot point of the brifter and check the handlebar for any scratches, bents or deformations (you probably need new bar tape anyway).