Ouch indeed. You strike close to home!

Sidney is the "easiest" to deal with. He was a Quartermaster with an infantry battalion - the 10th Sherwood Foresters. This battalion went out to France in July 1915 and served with 17th Division throughout the war. A kindly Sherwood Foresters researcher has forwarded me the War Diaries which I can pass on to you. They are 45mb in size so might take a while. PM me your e-mail address.
The basic outline of their service can be gleaned from the following:
http://www.1914-1918.net/notts.htmhttp://www.1914-1918.net/17div.htmThe Quartermaster was an "odd" appointment in an infantry battalion. He was an Honorary Officer (usually a Lieutenant or Captain), but did not have the King's Commission. Therefore a Honorary Captain could not give orders to a Commissioned Second Lieutenant or Lieutenant, but as far as the rank and file were concerned, he was an officer. The Quartermaster role (at least during the Great war) was nearly always filled by a long-serving Warrant Officer (Company Quartermaster Sergeant, Company Sergeant Major or Regimental Sergeant Major) who often had years of experience in the logistics of helping run his unit. The Quartermaster was of course in charge of obtaining stores and supplies for his battalion (usually in concert with the Transport Officer who was a Commissioned Officer).
The Quartermaster of an infantry battalion would not be a right in the front line soldier - Sidney's last shot in an anger was probably against an Afridi tribesman in 1897 - but he would have been part of the Headquarters Section, which would have put him just behind the front line most of the time, and sometimes in it. He basically organised all the food and stores coming in via Brigade HQ for distribution to the men. For example, in a battle situation this could mean he and his men would have to cross the old No Man's land to supply the troops who were holding German trenches.
Therefore, Quartermasters did get killed and wounded, though it was a (marginally) safer job than the infantrymen. As a result, many battalions had one officer who served through the entire war with the same battalion - the Quartermaster. In many ways, the Quartermasters were the nucleus of the battalion.
I've had a check through the war diary but Sidney Pearsall gets no mention - this is not unusual for a War Diary. They only usually mention even officers when the join, leave or get killed, and often the Quartermaster is there in the background getting on with his job - any mention of stores, re-equipping, supplies and dumps is probably him!
So, no Sidney did not miss the war... He saw the Somme, Arras, Passchendale, the German Spring Offensive of 1918 and the "Advance to Victory" from August 1918. Pretty much all of it!
T P H is a bit more difficult. I can find no trace of a medal card for him, so it is likely that he did not serve overseas, and was instead employed in the UK (including all of Ireland at that point). His appointments show him as an "Assistant Commissary Officer, and Honorary Lieutenant" and later an "O.E.O. 3rd Class" (Ordnance Executive Officer, 3rd Class). Both ranks are Honorary Officer ranks in the Army Ordnance Corps, equivalent to Lieutenant rank - much the same as Sidney...
The Army Ordance Corps was a branch of the Army, totally separate to the Infantry (in answer to your question in Post #1 !) , and which specialise in the supply of Ordnance (i.e. weapons, bullets, shells, etc. - anything to throw at the enemy, basically) to the Armies at the Front.
It is probably going to be difficult to pin down his job any further without his service records. You or your father shoould be able to secure a copy from the MOD.
Steve.