We then have the "Unit letter & No. (???); and, finally -
We have the name of the tank.
Where are these allocated to the tank and who applies them?
If I look at the pictures of three tanks knocked out during the Battle of Cambrai, Eclipse. II. has its name in three places, but no other markings.
The same applies to Eileen II, but Eve might (and I stress might) have the "Unit letter and No. on the starboard side, but no name on the front. Instead it has a drawing of two flags and the name on the port side.
War Department numbers are a misnomer, because they were allocated by the Tank Supply Department / Mechanical Warfare Supply Department / Mechanical Warfare Department (the department was renamed twice during the course of the war) of the Ministry of Munitions - so nothing whatever to do with the War Office (not War Department).
Unit letter and numbers are crew numbers, not tank numbers. However units became confused about this so it's not surprising we are similarly confused today. These were allocated by Battalions.
Names were probably chosen by crews, but equally probably required some sanction by senior officers. The formal system of giving tanks names beginning with the battalion letter was introduced after the Battle of Messines but before Third Ypres, probably mid-July 1917.
Yes, Some E Battalion tanks are marked with flags. Eve is not the only example. My personal theory, I have no proof, is that these ae "flag tanks" used by a section commander. This practice appears to be unique to E Battalion.
I very much doubt Eve has the crew number painted on the tank side. E Battalion at Cambrai just didn't do that - nor did G or H Battalions.
No, they are not there. This photo is of tanks that took part in the Battle of Cambrai between November and December 1917. The WRW flashes weren't introduced until Spring 1918. Therefore no tank at Cambrai had them.