IIRC there weren't many, if any 'Askari' battalions in the Western Desert in 1941/42. The Libyan divisions were virtually wiped out during Operation Compass and I don't think there were any Eritrean or Ethiopian units in North Africa, they were all concentrated in East Africa.
As for TO&Es, if memory serves the Askari battalions were differently organised to metropolitan, Libyan and Blackshirt battalions. I can't lay my hand on my copy of Command Decision which has good organisational detail but I would suggest you ask over on the CD Test of Battle Forum or just search the archive, there are some very knowledgable chaps including at least one Italian, with in depth knowledge of the subject. There's actually a whole thread on the composition of Askari brigades and battalions, which even includes the details of the cummerbund patterns and colours for each battalion. Well worth taking a look.
Leaving that to one side, pretty much everything was motorised in the WD. That didn't mean the Italians had sufficient MT to move everything at once but each unit was able to move by motor transport