Does v1 Segwit includes v0? My question is whether the version 1 of Segwit script will allow you to use version 0 scripts types inside of it. In other words, is version 1 superset of version 0? Please include reasoning behind the design decision.
If this question is about the recently published bip-taproot proposal, the answer is no, you cannot use a v0 script inside a v1 output.

The general reason for this is that some of the goals the BIP is trying to achieve are incompatible with v0 scripts. That on itself doesn't make it impossible to support v0 scripts, but reasoning about the benefits of the changes is much easier if you know there is no way to bypass them. Furthermore, the changes between (the initial set of) supported features in v1 and v0 is small:


No CHECKMULTISIG The OP_CHECKMULTISIG opcode is unnecessarily inefficient, as it sometimes needs to try multiple combinations of public keys with signatures. Furthermore, it is not compatible with batch validation (the ability to verify all signatures in a block faster than verifying them individually). As multisig constructions are very useful, a replacement opcode is available: OP_CHECKSIGADD, which increments a counter based on whether or not a signature check succeeded.
Schnorr instead of ECDSA All opcodes that take ECDSA signatures in v0 are changed to take Schnorr signatures instead. They're more efficient, but also ECDSA does not support batch validation, conflicting with one of bip-taproot's design goals.
Improved sighashing Some improvements are made to the hash that's signed in signatures to resolve a few long-standing issues. If there was a way to bypass these improvements it would be hard for other participants in a transaction to rely on these fixes being guaranteed.