Do I have to use the latest drivers for the FT232R and FT245R?

No.  The FT232R and FT245R are treated by the driver in the same way as the BM series devices, so any driver that worked with the BM series devices will work with the R series devices.  Some new functionality (e.g. EEPROM programming functions in the D2XX driver) may require the most recent drivers.  Any existing application code written for the AM or BM series devices should work with R series devices, requiring no or minimal changes (e.g. EEPROM structure).

Why does my FT232R/FT245R prompt me for drivers each time I connect a different device when my FT232BM/FT245BM did not?

Windows installs devices according to serial number.  If a device is connected with a new serial number, Windows sees the device as a new one and asks the user which drivers it should install for it.  It is likely that the BM series devices used either had no EEPROM fitted, the EEPROM was blank or they were programmed with the same serial number.  The R series devices have an internal EEPROM which is given a unique serial number during manufacture, so each R series device will need to be installed when it is connected.  There are ways to minimise the effect of this, such as using location IDs (see AN232B-07 Configuring FTDI's VCP Drivers to use Location ID's) which would install a device by it's physical location on the bus rather than according to it's serial number, allowing the same USB port to be used for devices of the same type with different serial numbers without re-installing the driver.

Do the FT232R and FT245R contain the MPSSE from the FT2232C device?

No.  For applications requiring the MPSSE (e.g. USB-JTAG, USB-I2C, USB-SPI, etc.) the FT2232C is currently the only device available. 

Are the FT232R and FT245R pin compatible with the BM series devices?

No.  This is due to external components (EEPROM, clock source and series resistors) being absorbed into the FT232R and FT245R devices.  The FT232R and FT245R are available in 28-pin SSOP and 32-pin QFN packages while the BM series devices were available in 32-pin LQFP and 32-pin QFN packages.

Are the FT232R and FT245R replacing previous generations of FTDI's ICs?

Although the new FT232R and FT245R chips are recommended for new designs, there are currently no plans to stop producing the two previous generations of these devices FT232BM/FT23BL/FT245BM/FT245BL (second generation) and FT8U232AM/FT8U245AM (first generation).