Can an USART peripheral works (in Synchronous mode) with only TX, RX and Clock lines. And without the other left handshake signals (CTS, RTS, DTR, DCD)?
Yes. Neither mode actually requires the handshake signals. The handshake signals are more of a convenience feature that can be used in each mode to determine when the devices are ready to do something. But they can easily be ignored (and often are).
The asynchronous mode requires only the rx and tx lines. Transmission rates are generally slower in asynchronous mode to ensure good performance. Since the clock signal is not provided, timing conditions must be recovered from the data signals - typically in the form of start and stop bits. These extra bits reduce the amount of useful information that can be transmitted in a given time period.
The synchronous mode requires only clock, rx, and tx lines. Transmission rates are generally faster in synchronous mode. Since the clock signal is provided as a reference, start and stop bits are usually not necessary.