On a day to day bases onboard a blue water ship you are the safety officer and wile on watch you will be the OICNW and/or in charge of cargo handling. When docking a lot of ships have the third mate filling out the bell book or assisting with the lines.
Onboard work boats (tugs and OSVs) it will be different.
On the PCTC I'm on the third mate stands the 8x12 watch with an AB.
Three mates needed for mooring and unmooring operations, the third mate is usually on the stern with a hand held radio in charge of the aft crew, two ABs and an oiler to run the winches. Some captains have the third on the EOT and the chief mate aft to spot the ramp.
Only two mates are needed for anchoring. Chief mate on the bow, third or second mate in the wheelhouse. If the second mate is on the bow then either C/M or third mate in the wheelhouse.
The third mate is also sometimes used when picking up a pilot depending on the time of day etc, either in the wheel house or at the pilot ladder.
Depends on the ship, and how many mates there are. On a standard 3 mate ship 8 hours of watch at sea/in port, 4 hours of OT for safety inspections, etc etc.. While entering/leavign ports, 3ms are usualyl on the stern for tying up. With the CM on the bow and 2nd Mate on the bridge. But like previously posted, be prepared to do any task, including dropping/heaving anchor. If you are on a tanker with 3 mates, expect to do a lot of 6 hour watches..
The specific tasks on the stern during mooring/unmooring operations would be to direct the crew laying out and preparing mooring lines, to make up/ let go a tug, run/let go the lines. Act as a lookout aft and give distances as required to the bridge.
In the wheel house specific tasks for the mate include: operates the EOT, ans phone, insure commands are followed properly, maintain a record (bell book or log).
In general the third mate should maintain situational awareness.