CSMA/CA - What happens if a faulty WiFi station does not silence itself when an RTS/CTS is received?
Will that affect the whole network?

All stations that now lose packets are affected. I.e. every station at which your malfunctioning's station is strong enough to damage received packets.

How this kind of an issue is handled?

Well, since that bad station won't always have something to send, it will probably not be handled at all: packets that collided will simply be re-sent, which the user only notices through increased latency and reduced throughput.
If that bad station continuously transmits, the its WiFi channel is simply blocked, which means the other stations notice that none of what they send ever gets acknowledged, and they'll leave that network eventually. An access point might use that knowledge and change channels, but that would assume it's configured to do that.

CSMA/CA is applicable only for the connected stations or all WiFi stations in the vicinty of the stations and APs?

Think about this logically: if you only forced connected stations to adhere to the rules of cooperation, everyone would suffer when a new station enters an area.
So, it doesn't only apply to connected stations. It also applies to unconnected stations. And even non-WiFi devices: in many countries (most) any ISM-band device with a maximum duty cycle above 1‰ typically needs to do &quot;listen before talk&quot;.