Is it possible and preferable to store my Bitcoins both in Counterwallet and duplicated locally on my pc/server for backup? What happens the other place when I spend from one place? Is it somehow synced?
Yes, you can have multiple wallets running concurrently with the same private keys - e.g. Armort, Bitcoin Core and Counterwallet. They are all synced by the data in the Blockchain. If you spend BTC from one wallet, you can’t spend the same BTC from another wallet (this is the double spend problem that Bitcoin inherently solves).
Yes you can pay for things with BTC directly from Counterwallet. It is a normal Bitcoin wallet, just with loads of additional Counterparty features too!
It is dangerous to use Bitcoin-Qt client that way (also, you need to run bitcoind and counterpartyd on the system), but it can be done. There’s no syncing issue, but there may be other issues, and it’s less safe too (as the wallet on your Bitcoin-Qt is exposed to risks). Best don’t try this if you’re not experienced with Bitcoin-Qt.