Yes, for example "I am getting better at English" is the same as "I am improving at English" :) "getting better" can also mean to recover from an illness, e.g. "hope you get better soon!" Usually you wouldn't use improve in this context.

Yes, it is. It can be 'get better' (intransitive) as in 'I rained in the morning, but the weather got better in the afternoon'. Or it can be 'make better' (transitive) as in 'The developers have improved the design of the product. It's better than the previous model.'