The King's Liverpool Regiment had nearly 50 battalions in WW1 but fortunately not all of them served in France & Flanders.
His date of arrival overseas may be a clue, if he was part of a unit, but he may have been part of a draft of reinforcements.
The nearest arrival date I can find for a unit is the 1/8th (Irish) Battalion of the King's who arrived in Boulogne on 3 May 1915. They were Territorials, but not all were Irish!
Someone with knowledge of numbering may be able to offer more.
The medal index card you quote shows the date of his entry into the war zone: 1st May, then his number in the King's Regiment, and later in the Labour Corps. This transfer often meant that a soldier had become unfit for front-line duties in the infantry from wounds or sickness. His rank was Private in both.
He was entitled to the usual three campaign medals - Victory [not Victoria], 1915 Star and British War Medal. If you have them the name, rank and number should be stamped in the rim. The other mumbo-jumbo is a reference to the Medal Rolls which are held at the National Archives at Kew. Sometime they will show more details, including a battalion number, but not always. They are not available online.
Class Z was the part of the Army Reserve to which he was allocated when demobilised.
Official photos of individual soldiers are very rare, and private ones are usually found among family papers, taken when about to leave for the front. Sometimes they appeared in local papers, especially in the earlier part of the war.
The 19th century newspapers online mostly go only to 1900, so not much help here.
A