with just the router and bit, will I be able to control the depth of the cut and align the router to make sure it stays centered in the wood? Would I need something like the plunge router attachment, referencing the router posted by Unstupid?
Both models Unstupid posted have a micro-fine depth adjustment. (highly desirable feature)
Ensure the bit is tightly secured in the collet before you begin cutting and check regularly to ensure it hasn't loosened.
Do test cuts on a scrap piece of MDF to align the depth before you start on the actual cab parts.
You can do almost everything you need to do with either a fixed base or plunge router.
Plunge routers can be locked in the down position, though sometimes that
doesn't work so good for t-molding bits.
Some routers even have swappable fixed and plunge bases like
this one.
BTW the video above shows him cutting clockwise -- he should be cutting counter-clockwise.
Scott