1)
No, your luggage allowance will not increase in that manner, so do not make the mistake of assuming you can now check a third bag without being hit with an even higher fee for that third bag. Business class only ever comes with two free baggage items; you will have to pay for any third. Bear in mind that the fee for a chargeable third bag is likely far higher than the fee you paid for a 2nd bag in Economy, so you are likely to be required to pay considerably more money again, should you now wish to check a third bag.
It is not clear that you would be due a refund for the extra baggage fee already paid if you buy this upgrade and travel only with 2 bags; it is likely that the airline will say that this fee has already been factored in to the upgrade offer you received.
2)
No, if taking an upgrade offer at checkin for only part of the journey, the increased luggage allowance would only count for the upgraded leg and not for any feeder flights either before or after the upgraded sector, and neither would you have any increased baggage limit on any return/further sectors on the ticket that have yet to be flown.
My advice - do not bring a third bag just because you are considering taking an upgrade offer.