Can I connect a Belkin Thunderbolt 3 dock to a Thunderbolt 2 MacBook Pro 2013 with this adapter? The dock will have a 4K screen, keyboard etc.

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    This should be possible. Thunderbolt 2 allows 20 Gbps. A 4K 60 Hz display takes about 16 Gbps over Thunderbolt. This leaves 4 Gbps for everything else (for transmit, and up to 20 Gbps for receive).

    That will reduce the write speed of a USB 3.0 device (normally 5 Gbps) or a USB 3.1 gen 2 device connected to one of the Thunderbolt 3 ports (normally 10 Gbps). The read speed should not be affected.

    You may wish to connect the 4K display directly to your MacBook Pro 2013 instead of the dock so that the write speeds aren't affected.

    According to apple technical support document HT206587 "Using 4K displays, 5K displays, and Ultra HD TVs with your Mac", the MacBook Pro (Retina, 15-inch, Late 2013) doesn't support 4K at 60 Hz unless it is a MST display?

    You did not specify if your MacBook Pro is 13 inch or 15 inch, or if it is early 2013 or late 2013. Your MacBook Pro may be limited to 4K at 30 Hz. In that case, the display will use about 8 Gbps over Thunderbolt, allowing 12 Gbps for transmitting.

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