What is crypto-proof UTXO set? There are many opinions about getting the actual UTXO set of blockchain.

Do you think that this important database must be based on data which saved on disk and never modified after, only appended?

I mean that only in this case of history based calculations, provide high level of node security.

Is it true that UTXO set which has the full node for to verifying transactions calculates only from recorded on disk information? Or it is a compressed data received from the network?
In bitcoin, the UTXO set is derived independently by every full node by processing and verifying transactions in blocks. It is then stored on disk until it is spent. When a UTXO gets spent, it gets removed from the UTXO set so the UTXO set is definitely modifiable. When full nodes verify transactions, they also use the UTXO set on disk to check the validity of each transaction. Full nodes have to make sure that UTXOs consumed as inputs in a transaction actually do exist in the UTXO set, otherwise a transaction will be invalid.