Sword in the Heavens: German Ground Based Defences 1918 - 1945 by Edward Westermann answers some of your questions:
In September 1941 the flak arm had 967 heavy and 752 light batteries. Over the next month another 5 batteries of 105 mm guns were added, along with 4 batteries of 37mm, 5 of 20mm and 2 of 40mm guns.
At the same time 49 new "barrier fire" batteries were added. These batteries had the guns but little in the way of optical and radar equipment, and had to fire barrages rather than aimed fire.
The standard size of a battery at this time was 4 guns, although the Luftwaffe had just begun to experiment with larger batteries.
As to RAF sorties, you can find a lot of figures at
BC Main Page
Night sorties, second half of 1941:
Aug - 3354
Sep - 2621
Oct - 2501
Nov - 1713
Dec - 1411
Westermann also gives details of the number of aircraft shot down by Flak over Germany and Western Europe in the second half of 1941:
Month - Day - Night
Sep - 115 - 45
Oct - 52 - 47
Nov - 33 - 41
Dec - 16 - 33
Bear in mind these are the German figures. They will include aircraft from other RAF commands, in particular the fighters that were flying a lot of operations over France and the Low Countries.
Westermann goes on to say flak brought down 242 aircraft in the west at night in the last 6 months of 1941. Night fighters accounted for 421 in the whole of 1941.
The flak forces accounted for another 1,325 aircraft between October and December, mostly in the East.
Westermann also discusses the economics of the flak arm. He has a table showing the percentage of the entire armed forces weapon and ammunition budgets going on flak in 1941:
1st Quarter - Guns 15% - Ammo 18%
2nd Quarter - Guns 17% - Ammo 27%
3rd Quarter - Guns 19% - Ammo 34%
4th Quarter - Guns 24% - Ammo 35%
He then comments:
The devotion of over one-third of the Wehrmacht's entire ammunition budget to anti-aircraft munitions in the last two quarters of 1941 once again highlighted the importance placed by Hitler on the strengthening of the Reich's ground-based air defences. Several historians have questioned the large-scale diversion of resources to flak ammunition and flak equipment. In turn, many have argued that these resources would have been better spent on building more fighters. It is important to note, however, that the United States Strategic Bombing Survey found that "since earlier limitation of output was largely the result of deliberately restricted demand, it cannot be said that the investment in antiaircraft prior to 1943 represents a cost in terms of other weapons and ammunition." In other words, the opportunity costs associated with expanding the flak arm in the first three years of the war do not appear to have negatively impacted the overall German war economy prior to 1943. Furthermore, the increased production of fighters also entailed numerous hidden resource costs including expanded pilot training programs, increased fuel demands, and the necessity for more air bases, maintenance depots, and supporting aviation infrastructure. In short, the calculus of air defense did not allow itself to be reduced to simple binomial equation.