"On April 2nd, 1941, the Heavy Battery Number 833 of the new battalion was formed, each battery having two howitzers. Orders arrived to be combat-ready by May 1st, 1941, in time for Operation Barbarossa. Initially, a single battery was deployed against the Soviet fortress at Brest-Litovsk while the second battery was ordered to support the attack by the 4th Army of Army Group Center against the Brest Fortress. Being their first deployment in combat, not much is known of the Battery's operations except that IV Army Corps reported in late June that the battery itself was no longer operational due to technical firing problems. The second battery's weapons had some assembly problems along with an electrical firing mechanism issue. The size of the shells required some individual production in a few armories so each shell was made as non-standard ammunition, effectively not to precise calibrations. The Karl-Gerat's second battery's introduction to a combat situation fired 31 of their 36 rounds at the fort by June 24th. The battery was ordered home by Army Group Center where the battalion reformed with eight 21-cm Morser 18 howitzers in August of 1941. Karl did not break the fortifications as expected due to her aiming problems that effectively reduced concentrated fire and added to her already existing mechanical issues.
The Karl was used for the shelling of Sevastopol by the Germans in 1941 and into 1942. Sevastopol was under siege by the German Army and was corralled for 250 days before the city fell in July of 1942. Heavy Artillery Battalion 833 was ordered to form a Battery comprised of three Karl howitzers and support vehicles in February of 1942. Support forces for the Karl weapons traveled to Sevastopol to survey firing sites in preparation for the attack on the city to be scheduled for the early summer. The support teams dug these future secret firing positions so the battalion could quickly move into the firing positions with little lost time. Each pit for a Karl was 15 meters (49 ft) long, 10 meters (33 ft) wide and 3 meters (9.8 ft) deep and had to be dug for each howitzer to minimize Soviet artillery return fire"
This was cut-n-pasted from;
http://www.militaryfactory.com/armor/detail.asp?armor_id=297
Thor in firing position, Sevastopol.
A very nice close-up of Thor;
As best as I can tell, this looks like overall Panzergrau with a brushed-on pattern of mud.
I have a bit more info on "Thor's" actions at Sevastopol against Fort Maxim Gorky.
If you read German or feel like going big with Google Translate, there's more here;
http://www.forum-der-wehrmacht.de/index.php/Thread/27868-Geschütz-Loki-Mörser-„Karl“-Gerät-040-041/
And of course there is always Panzer Tracts;
https://www.perthmilitarymodelling.com/reviews/books/panzertracks/pt_karl.htm
That's about all I've got on early actions while they were still grey. I have more info in a couple of books, but this deals mainly with technical aspects of the weapon. It's hard to tell which gun was where, and I don't know how much more is known about their early use, 
at least at the moment...