Responding to "Will these continue to cause flares in the same predictable manner, and about as regularly?" (SEE EDIT at foot of answer) the original Iridium satellites and the Iridium Next generation are of different physical design, see the following photos from these sources:
Original (Earth is to the right as this photo is oriented)
NEXT (despite the artists image of the planet, this satellite points to Earth in the lower left foreground)
The current Iridium satellites aren't unique in producing strong flares though I think it would be a coincidence, give the huge variety of satellite designs, if the replacement constellation was similar.
Four the part of your question "If not, how much longer can we expect to see flares from the existing constellation " I think this will be driven a) from the deployment schedule of the new generation adhering to plans and b) the remaining propellant of the older satellites and how they are managed after retirement from the operating fleet.
EDIT: This link is a bit more direct:
Iridium flares will be gone in a few years when we launch our next generation of satellites, Iridium NEXT. Due to design improvements, our new satellites unfortunately are not expected to create flares, so enjoy Iridium flares while you can. The Iridium NEXT satellites will launch in 2015-2017 and will have only one antenna instead of three. The new antennas offer increased data speed and capacity but are angled differently than the current design, and are therefore not likely to reflect much sunlight.