the new residence requirement says that in 2019 you must have stayed in Denmark for at least 5 of the last 12 years.
And since you came here in 2013 you meet the new residence requirement.
From 2021 the new residence requirement will be 7 out of the last 12 years, but at that time you will have stayed here 8 years, so you will have no problem with the new requirement.
So you should not stop paying to A-kasse, only because of the new requirement.
And that leads me to your next question..it is very important that non-EU citizens consider if they have the right work permit. A work-permit that are limited to a specific job, will not give you right to work in Denmark if you lose that job, and you must be able to work full-time to be eligible for Unemployment benefits ("dagpenge").
So whether you can get benefits in july 2019 or not, depends on your work-permit. I assume that you in july have been member of an A-kasse for 1 year, and also meet the income-requirement.
You are not allowed to work in Denmark in the 6 months jobsearch period. However if you have applied for a new work-permit before the current expires you are also allowed to work in Denmark in the "waiting-period", while your case is being processed.
We can not give advice about work-permit - you must check with others what the rules says specifically about this (whether you are allowed to work in Denmark or not).
So with the current visa you have, it seems for me that you must consider if it is worth paying for A-kasse. Only if you think you can work in Denmark also after your current job, it will be relevant to pay for A-kasse membership.