1. Any length is ok, both TX and RX as long as it fulfills your needed system functionality. λ/4 is a practical length due to its impedance is reasonable easy to match against common types of TX & RX impedances. Shorter monopole antenna then λ/4 will reduce antenna efficiency and radiation resistance will be lower, causing reflection losses if not compensated for.
Reduced overall antenna efficiency can be compensated with higher TX power level. For RX is it a bit more complicated as added amplification to compensate for reduced input level also adds noise from internals of amplifier which reduces signal noise ratio. Antenna length do also effect radiation pattern, especially when monopole antenna length is longer then λ/4.
2. Can be a lot of reasons. An antenna that must be heavily impedance matched to fit system-impedance do often become narrow in its usable frequency range due to properties of matching elements and for real low radiation impedance do resistive losses in non ideal matching components become an important factor.