Regarding the regular army for 1810-1812, unless you are in Britain and able to access the Public Record Office, I do not think you will get full answers to your questions, especially where the regiments [regular and militia] were actually stationed.
If you have access to the House of Commons Journals you will find information on strengths and establishments, but not on actual numbers per regiment. I am not certain if by that time, they still indicated which battalions/regiments were serving overseas.
I can help with an incomplete list of General Officers commanding the districts 1810-1812. England was divided into 15 Districts with stations under command [reduced to 14 by 1812]. For example the Yorkshire District had two stations, Beverley and Hull.
Some of the District Commanders who held commands during the period in different years were:
Commander-in-Chief General Sir Davis Dundas, then Field Marshal HRH Duke of York
Lieutenant General Francis Dundas,
Lieutenant General Sir Charles Green
Lieutenant General Richard Vyse,
Lieutenant General John Pitt, Earl of Chatham
Lieutenant General Oliver Nicolls,
Lieutenant General Lord Charles Somerset
Lieutenant General Richard Lambart, Earl of Cavan.
Major General Arthur Whetham
Major General James Taylor
Lieutenant General Richard England
Lieutenant General Banastre Tarleton
Major General George Warde
Major General John Oswald
Major General Joseph Champagne,
Major General John Robinson
General HRH Duke of Cambridge,
Lieutenant General Sir Harry Burrard
Lieutenant General Henry Pigot
Major General Craufurd
Major General William Dyott
Lieutenant General Baldwin Leighton
Lieutenant General George Don
Lieutenant General Sir John Doyle
Lieutenant General Thomas Maitland
Scotland [referred to as North Britain] had six stations:
Commander of the Forces: Lieutenant General William Cathcart, Viscount Cathcart
Lieutenant General Alexander Campbell
Lieutenant General James St Clair, Earl of Rosslyn
Ireland was divided into five districts:
Commander of the Forces: General Charles Stanhope, Earl of Harrington then Sir John Hope.
Major General Archibald Campbell
Lieutenant General Sir Charles Asgill
Lieutenant General Henry Wynyard
Major General Edward. Dunne
Major General Sir James Affleck
Lieutenant General John Floyd
During this period 1810-1812, the home based battalions, mostly all 2nd battalions, would be used as a depot to reinforce the overseas battalion, so yes there would probably be a higher percentage of new or partially trained recruits there. For the infantry regiments which were serving with Wellington's army, based on returns for September to November 1812, there were 31,163 men with the army, 18,151 sick with the army, 4,486 on passage to the army and another 6,482 men available in England for immediate service. For the cavalry there were 5,751 with the army, 1,436 sick with the army, on passage were 712 men and ready for immediate service in England another 696 men.
The single battalion regiments and those which had both overseas would have only small depots in England to train reinforcements.
The militia would be assigned to the different districts and if I recall correctly the militia would serve outside of their county and even farther afield [that is Irish militia in England and English and Scots militia in Ireland]
Hope that this helps somewhat.