1) I can understand that downgrading regulars into militia reduces both cost and effectiveness, but with some investment can the militia be upgraded back to regulars?
Yes, it takes some times until the effectiveness increase.
2) In terms of one-on-one effectiveness, are militia stronger or equal to rebel scum?
Difficult question: a militia lead by a powerful general (Frederic II or Gustav-Adolf) will be a very efficiency army.
On contrary, the same army lead by general Anyone-can-do-the-stuff can go directly from defeat to defeat.
In the same way, there is a great difference between rebel scum. Some of them are very strong (rebellious armies in civil war are tremendous adversory, sometimes lead by yours best generals

)
I would say (it is only my appreciation) that the worst milicia in general case is at the same level than the average rebel scum.
3) Do you have to convert your entire military into militia or can you designate only certain armies?
Your entire army has the same level (but each detachment has it own moral depending of their victories, defeat, rest state).
4) Are ships, cavalry artillery also converted into militia or does this only affect infantry?
Everything can be converted into militia, but land forces are separated from fleet: you can choose to have the best navy and the worst army.