Since Native Segwit addresses (Bech32) use lower fee as compared to Segwit addresses, then would it make sense to ask the online merchants to provide a Native Segwit address for accepting payment instead of a Segwit address?
You can try but it does have some downside. There are several services that are still incompatible with Bech32 addresses (god knows why) so it'll be wise to use a P2SH Segwit Address for better compatibility. I presume some of the services actually offer both and toggles between the types of addresses which would be great.
either "1" or "3" which indicates legacy and segwit addresses respectively.
Addresses starting with 3 merely indicates that it is a P2SH (Pay-to-Script-Hash) address, it does not necessarily contains a witness script wrapped inside.
This also explains the fact that I often received a "high network fee" warning from the Bitpay app while making some of these payments.
Probably not. There shouldn't be any noticeable difference when you're paying to a legacy or a segwit address.
Interesting, in that case, it would make sense to have Native Segwit addresses in Bitpay wallet to save on the fee I am spending when I send transaction to another address?
Yes.