1) It's either that or machine something which fastens to the MDF to hold them, but since presumably the MDF is a temporay measure I'd stick with just pressing them into the MDF. Press them up against a shoulder (see 3).
2) Yes
3) Machine the shaft such that the bearings are a sliding fit, so they just fit on with no play. You shouldn't press them on since the bearings need to be preloaded, which isn't possible if the inner and outer rings are all constrained. Put a thread on the shaft and a nut to tighten and pre-load the bearings against a shoulder on the shaft. A fine thread and two nuts to lock against each other would work well, or a locknut/washer and grooves in the shaft but that's a bit OTT.