yes you should wait until the 3 months notice period has expired. You are not regarded as unemployed as long as you receive salary from your employer.
If you apply for "dagpenge" after 1st February 2020, there is no residence requirement. It is the date you apply that is crucial in relation to the residence requirement, not the date you stopped at work.
"The best" A-kasse and Union for english speakers is depending on your needs. If you have a business-related long higher education and still plan to work within the privat sector, then I think there are no other better solutions than "CA A-kasse". In CA A-kasse you can also buy "CA Advokathjælp" (legal help/Union) for DKK 50 per month.
I would think that CA A-kasse are able to guide you in english.
Anders Weber, www.a-kasser.dk