> Can my company file a greencard application on my behalf as soon as I enter the US
Not a lawyer but having been there... please do this. That goes for anyone else reading this as well. There isn't really a waiting period as far as I am aware. Companies take their sweet time because there's no incentive to do otherwise.
Even if you are not from the two backlogged countries, a successful application will most likely take years to complete. In the mean time, you would be dealing with the stress of the concurrent L1 or H1 extensions, which have become much harder lately, are RFE-prone and sometimes you get crappy 1 year extensions.
Another thing to note, specifically for L1: you need to prove you have company-specific skills that can't be found elsewhere, which is why they need to transfer you. That's different from H1, where you need to prove you have skills that can't be easily found in the workforce, but they don't need any company-specific knowledge.
But, for the PERM process, you have to prove that you have skills that are difficult to find in the broader US labor pool - and generally, that you had those skills before you were employed by the company (because, if you didn't have them, it means that they could hire a US worker without those skills and train them, just as they have 'trained' you). In fact, they will ask how long would it take to train a worker on your 'skill' level.
So you can see how the two skill sets (for L1 and PERM) don't overlap and how complicated the application can get. Another reason why you should start early.