Marketing Developer Platform - Usage of Test Organizations

Do I need to get approved first for the marketing developer API Access to post to test organizations?

Marketing Developer Platform - Usage of Test Organizations

Do I need to get approved first for the marketing developer API Access to post to test organizations?

Yes, Now I got my linkedin approved for "Marketing Developer Platform" Product under Products in my app dashboard. You can apply to the same by "Add Products"

Or is there another way for accessing marketing API without approval from linkedin?

For now, no other way to get around this. Comment below if you know.

I have applied on April 26, they said that it might take around 90 days to get it approved. Is somebody got approved lesser time than 90 days?

They got me approved within 20 days which very decent review time.

To migrate my users to v2 API, do I need to force my users to reauthenticate with OAuth with the new scope?

On May 1st, linkedin is going to deprecate its V1 API. So do I need to get my users reauthenticate again with the company permissions once I get approved?

No, your old API tokens will be valid till their expiry time. After that, your users have to reauthenticate to use your app. Make sure that linkedin API version endpoints are active. 

How to apply for programmatic refresh tokens API access, does marketing developer platform includes it or I have to apply it separately? If so please post the link for it.

No, for "programmatic refresh tokens API access" you have to apply separately.