Air carrier First Officer reported non-standard communications and procedures during pushback with the ramp agents; which resulted in uncontrolled aircraft movement when the tow bar was disconnected. The Captain set the brakes and no injuries or damage occurred.

Narrative

A series of distractions along with our ground crew not following SOP caused a small mishap during the pushback with the ground crew upon disconnecting the tug. The ground crew firstly wanted to commence the push while a cargo door was open. The actual mishap was when they disconnected after the push (with one engine running. They never used proper phraseology pushback complete set brake". The aircraft's parking brake had not been set due to this and due to other distractions happening during that time. The distraction at hand specifically in this moment was a message regarding needing a lavatory service. The purser had to be contacted to determine if we needed to shut down the engine and go back to the gate for the service; which was not needed. After this; we began to start engine 2 at which point the aircraft started rolling forward. The captain caught the error; stopped; and set the brake avoiding what could have been a bad situation.It seemed that there was one distraction after another from the beginning this morning. First a maintenance issue followed by performance issues with errors from our heavy takeoff weight. The ramp had a hot mic and we could hear shouting and talking even after asking them to disable hot mic. Next was the attempt to push with cargo door open followed by the lavatory servicing message. It is important to mention that these distractions are not uncommon; but they become very challenging to manage and mitigate threats after a minimum rest layover with an early departure time. I never felt fatigued enough to warrant a fatigue call and was sure I was fit for duty; however; minimum rest does have a negative impact to safety."