Where and when are cryptographic algorithms used (During Boot , Playing games, typing, Internet

To my knowledge and experience:

During boot: yes - e.g. Secure Boot uses cryptographic signatures to verify that the system firmware and OS have not been tampered with. Or Disk encryption (e.g., BitLocker, LUKS) ensures data confidentiality before unlocking the drive.

Playing games: probably only if its online 

Typing: Some wireless keyboards might use encryption, but not always

Internet: Yes, everything you connect to via https uses encryption (TLS) and there are cryptographic algorithms to secure authentication (password hashing, two-factor authentication)

I don't know that much about the TPM 2.0, but in most of the cases new versions offer support for newer algorithms and have a more secured/improved architecture. Over the years, cryptographic algorithms need to generate longer keys to maintain their security, because the computing power in general is improving so fast. And there are new algorithms like ECDSA which are in general more efficient to save time.
But ofc new things might have vulnerabilities, especially in the beginning. Nevertheless I would say they are necessary.