It's always been "a lot" and "all right", even though "alot" and "alright" are very very common errors--and I know some people who learned them in school (!)
"Awhile" versus "a while" is more complicated. Usually, it's correct to use "awhile" only as a simple preposition-- "I was there awhile" or "Stay awhile." If you have a more complex clause, it's really better to use "a while"-- "After a while, every donut tastes the same to you."
Note: What the Merriam-Webster is saying on this is the same as what I'm saying, only in dictionary-speak. "A while" is always correct; "awhile" is correct as a simple descriptive preposition.