Since the "ownership" of bitcoins is essentially tied to having knowledge of the private key for the address they are stored against, it's very difficult to prove that nobody knows the private key.

However, it is possible to send bitcoins to an address with no private key, essentially destroying them. Details here: How to generate a valid bitcoin address for destroying bitcoins?

To answer your 3 questions:

1. There is no accepted definition of a lost bitcoin that I'm aware of, and it's unlikely there will be one because of the difficulty in providing proof.
2. The network doesn't know whether the private key for some bitcoins have been lost.
3. The value is based on market demand. From this perspective, bitcoins that aren't currently offered for sale are the same as lost coins until they are offered up for sale, at which point the price is affected. Though individuals do have the capability to prove they own bitcoins without offering them for sale, they rarely do so, so you'll probably never be able to find out much about which unspent bitcoins are still being quietly held and which are lost.

Since the "ownership" of bitcoins is essentially tied to having knowledge of the private key for the address they are stored against, it's very difficult to prove that nobody knows the private key.

However, it is possible to send bitcoins to an address with no private key, essentially destroying them. Details here: How to generate a valid bitcoin address for destroying bitcoins?

To answer your 3 questions:

1. There is no accepted definition of a lost bitcoin that I'm aware of, and it's unlikely there will be one because of the difficulty in providing proof.
2. The network doesn't know whether the private key for some bitcoins have been lost.
3. The value is based on market demand. From this perspective, bitcoins that aren't currently offered for sale are the same as lost coins until they are offered up for sale, at which point the price is affected. Though individuals do have the capability to prove they own bitcoins without offering them for sale, they rarely do so, so you'll probably never be able to find out much about which unspent bitcoins are still being quietly held and which are lost.