If you have not already done so, I recommend reading Medieval Hand Stitching – Basic Stitches (Start Here). It describes what supplies you will need for hand stitching medieval garb, how to start and end your thread, and the basic stitches upon which most other stitches are based. This tutorial describes how to hand sew buttonholes, especially for medieval reenactment.
Medieval buttonholes were cut before being sewn. To stabilize them while working some sort of clear glue, like fish glue, was applied to the buttonhole area before cutting and sewing. Fray check is the easiest modern option. Mark your buttonhole, apply glue to the mark and allow it to dry, cut your buttonhole, and then stitch.
I am not aware of any medieval garments that used a running stitch to border buttonholes, but it can be handy to clearly mark the area and to help prevent fraying of the fabric while stitching. If you choose to take this step, either a running stitch or a back stitch can be used. I would apply this stitch first, before gluing and cutting the buttonhole. It should be completely covered by buttonhole stitch when the buttonhole is finished, and therefor not visible on the finished garment.
Use a buttonhole stitch, stitching towards the buttonhole opening, and placing your stitches close together.
Medieval buttonholes did not have any stitching at the terminal ends. The only stitches are along the sides.
For later period or modern tailored buttonholes, definitely take the time to running stitch or backstitch around the buttonhole. Glue and cut as with medieval buttonholes. Then stitch around the cut on all sides, long and short. For tailored coats, it is appropriate to buttonhole stitch a rounded keyhole-like shape at the garment edge side of the buttonhole. You can use a plain buttonhole stitch, or any of the other myriad variations on buttonhole stitch, depending upon how you want the finished buttonhole to appear.