No you dont have to register with MOE

Yes, there are problems with then entering children into the school system here if they have been homeschooled, as MOE doesnt recognise HS at all.

You will need to check with them, but when I inquired a few years back, they wouldnt touch HS kids at all. (in fact, I believe you cant be "out of the school system" for more than 6 or 8 weeks without causing big problems)
They also need transfer certificates etc from other schools to enrol.

College/Uni may be different now...but I doubt it.
Most schools have children sit entrance exams here anyway, so possibly, an entrance exam may be suffient..but I would check 1st.

What ages are your children?