For a long time mirrorless cameras had terrible battery life and no phase detect autofocus.

A DSLR doesn't need to keep a screen powered all the time you're looking through it. Improvements in the battery life of MILCs has made that non-issue for most people though.

There are two types of autofocus systems that cameras use, phase detect and contrast detect. Contrast detect is what all point and shoot style cameras (and early mirrorless interchangeable lens cameras) used where they identified contrast of vertical edges and used that to determine if an image was in focus, this is a slow process as the camera needs to figure out what direction to move the focus and check the image at multiple focus positions to compare until the camera determined it was in focus.

DSLRs have a separate sensor up near the viewfinder that does phase detect autofocus, this is different in that it can determine how far the lens is from being in focus and go directly there which makes it significantly faster which was crucial in many professional applications.

Eventually sensor manufacturers figured out how to integrate a version of phase detect into the pixels of the sensor itself. While these didn't perform as accurately as the separate phase detect arrays in DSLRs, they had an extra trick where they were able to create a hybrid autofocus system that used the phase detect array to get close and then contrast detect to get accurate focus. It was only once this was implemented on mirrorless cameras that the advantages of SLR disappeared.

Yeah, as long as the camera and lens have same lens mount.

Edit: or an adapter I guess.

Yes they can. the technical difference is only in how the viewfinder works.