My question is similar to this but not quite. Why do mirrorless cameras have slower and less accurate auto-focus in general when compared to their DSLR competitors?
There seems to be some confusion as to what you are asking. This answer takes under assumption that you mean to compare AF in mirrorless cameras with through-the-OVF autofocus of DSLRs. In Live-View, both types of cameras behave the same and could theoretically perform identically. They do not in practice because components and processing are not optimized with the same priorities.
Wrong. Mirrorless camera focus is even more accurate than DSLR autofocus. Contrast-Detect AF used in mirrorless cameras is handled by a feedback loop reading the sensor and adjusting the lens and so has no alignment issues. Unlike the autofocus system used in DSLRs, it cannot suffer from back or front focusing issues.
Phase-Detect AF used in DSLRs is generally faster since knows which direction it needs to adjust focus and does not need to hunt. Panasonic is trying to achieve similar results by using Depth-From-Defocus to determine the amount of adjustment need to autofocus using Contrast-Detect and close the gap. To some extent, they have managed. The difference between Contrast-Detect and Phase-Detect autofocus in terms of speed is now very small.
Both types of systems are equally sensitive. Top DSLRs and mirrorless cameras can autofocus now down to -4 EV which is very very dark.
Many manufacturers of mirrorless cameras now use on-sensor Phase-Detect. While they use the same principal as DSLR Phase-Detect AF, they have more limitations since these are built-into the sensor while DSLRs uses separate specialized units.
Being able to read the sensor while performing autofocus is a significant advantage for mirrorless cameras which can perform Face-Detect, Eye-Detect and other features which are impossible with the Phase-Detect units used by DSLRs, even though when Live-View is engaged, a DSLR behaves the same as a mirrorless and indeed can (but does not always) have the same features as a mirrorless.
Of course both camps are also fighting to improve their systems and we have more and more choice to balance feature, sensitivity, accuracy and speed than ever before.