Does the bitcoin testnet3 network support Segwit and OP_CSV?
Yes, testnet3 supports SegWit as of block 834624.
Yes, testnet3 supports both OP_CSV and OP_CLTV.


You might also be interested to know that there is a segnet test network on another port number. See https://bitcointalk.org/index.php?topic=1453392.0 for more details.