Why the second instance (private) did'nt get Internet connection?

Private instance didn’t get internet because its subnet route table isn’t linked to NAT Gateway.

What is the purpose of Allocating Elastic IP while creating NAT gateway?

Elastic IP is gets a fixed public IP. This makes internet traffic routing stable and reachable without changing IPs.

How NAT gateway differs from Internet Gateway?

Internet Gateway lets public subnets talk directly with the internet .
NAT Gateway lets private subnets go out to the internet

Why NAT needs high availability?

High availability ensures continuous connectivity without downtime.

Is there any possible ways to make private subnet to connect to internet?

Private subnet can connect to internet using NAT Gateway