2a) Yes, you can continue to be member of an A-kasse, all though you will not be able to get dagpenge. It could be a good thing to continue your membership, especially if you find a new job relatively quick. By keeping your membership, you will not need to start over again with 1 year membership, and you can also use your income already earned to meet the income-requirement. However be aware that you can only count maximum of DKK 19,448 per month.
All A-kasser offer discount for unemployed members, if you have been in a 2 year period with dagpenge, which is now expired, and you have not (yet) earned a new 2-year period.
But in your situation I have not heard about that it should be possible to get discount.
2b) All work/income during the last 3 years up to the date you apply for dagpenge, can be included to meet the income-requirement. Only requirement is that you were member of an A-kasse, when you earned the income. Even if you have a break in A-kasse membership, then you can still count all work/income for last 3 years, from periods where you have been member. So you do NOT need to have continuous/uninterrupted 12 months of work. However be aware that you can only count maximum of DKK 19,448 per month, regardless of how much you earn. So you can never meet the income-requirement with less than 12 months of work. 12x19,448=233,376 (the income-requirement in 2019).
And you are right, the continuous part refers only to the A-kasse membership period of 1 year, up to the date you apply for dagpenge.
3) I think you can find the answer of this in my comments above. If you stop your A-kasse membership then you will lose the membership-periods, but not the income/work earned during previous A-kasse membership periods (3 years back in time). So if you stop A-kasse membership, you will always need to be member again for at least 1 year, before you can apply for dagpenge.