Is it possible to exploit the use of the same private keys on different crypto coins? Suppose someone has single private key, and from that private key he made a public key for Bitcoin, Ethereum, Dogecoin, etc. - is it possible to figure out the private key having those multiple public keys?
It depends on how many times that private key has been used and whether the attacker has access to a side channel. Practically speaking if the attacker does not have access to a side channel and does not know a vulnerability in the secp256k1 curve (there are no known ones), then no, it's not practically possible to figure out the private key.