"Next Monday, we .......research as usual."
a- do b- are doing
I think "do" because "as usual" refers to a timetable.
Which is better depends on the surrounding context. In a normal case, where what you are doing next Monday has not been scheduled and set in stone, so to speak, long in advance, "are doing" works much better:
- Next Monday we are doing research as usual.
You would use the simple present, however, if instead of "next Monday" you had "on Mondays." Then the simple present would indicate a weekly habit. But, as I indicated above, in an extremely unusual context it could work in your sentence.
- On Mondays we do research as usual.