In my system, multiple RTD measurements need to be done with MAX31865, how can I handle that using multiplexer such as MAX14778?

You may use multiple RTD devices with our MAX14778 beyond the rails multiplexer as the resistance of the mux is only about 1.5 Ohms but is very flat and matched between channels. This way an initial calibration (in your micro) will ensure that multiple PT100/1000s track each other exactly. This has been proven in an aviation application where multiple RTDs were used on a wing control surface. A typical configuration would be:

Instead of connecting VN to EP of the MAX14778, can I connect VN and EP directly to GND?

Q: Instead of connecting VN to EP of the MAX14778, can I connect VN and EP directly to GND? A: No, EP must be either connected to VN, or left unconnected. EP must never be connected to ground. VN is the output of the negative charge pump and the only allowed connection is a 0.1uF capacitor to ground. EP is connected to the die substrate with conductive epoxy. The substrate must be biased at the most negative voltage potential which is why we recommend connection to VN, but the substrate will self-bias to VN if the EP is left unconnected. The EP also provides a low thermal resistance path from the die to the board. The package thermal characteristics follow JEDEC specification JESD51-7 which assumes that the EP is soldered to a copper pad, therefore, for best thermal performance EP should be connected to an isolated copper pad and to VN. The actual dissipated power in the MAX14778 with a full 300mA rated current through each of the two multiplexer channels will be 2 x ((0.3A)^2 x 1.5 Ohms) = 540mW which is much lower than the package specified continuous power dissipation capability of 2666.7mW.

Can I cascade multiple MAX14778 chips to implement a 1:16 multiplexer for RS-232

Question: I need a 1:16 multiplexer for an RS-232 signal. Can I use a MAX14778 and connect another MAX14778 on each output to create a 1:16 Mux? Answer: Yes, you can cascade multiple MAX14778 devices to create larger multiplexer configurations.

Is there a Breakout/Adapter Board for the Max14778 IC for use with a Breadboard?

The IPC0012 : https://www.digikey.es/product-detail/en/chip-quik-inc/IPC0012/IPC0012-ND/5014803 board available from DigiKey matches the 20 pin TQFN-CU land pattern and has traces to 0.100" pitch pins that fit in breadboard/standard spaced headers.