For example, the 747 Wikipedia page says that 1500 747s were produced (that number seems quite low!).
Any indication of how many large, commercial aircraft there are, and what the breakdown between passenger and cargo is?
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As mentioned this question is hard to answer very accurately without better access to data. It also depends on your definition of "large, commercial aircraft." Including all wide-body aircraft would be a large number of planes, and the totals are constantly changing as aircraft are delivered and retired.
I looked at the widebody models on the first linked site, sorting by status to find how many are in "Active" service. This does not include "Stored" aircraft which will be either put into service or not.
The models below total: 1026 cargo, 3825 passenger. This means about 21% of the 4851 widebodies are cargo aircraft, and widebodies constitute about 24% of the worldwide commercial aircraft fleet of 20000 aircraft.
There are 652 aircraft in service of 1500 delivered to date. Of these, about 289 are cargo aircraft. Excluding 28 aircraft operated by militaries or other organizations, this leaves about 335 passenger aircraft.
There are 69 aircraft in service of 446 delivered. This does not include aircraft in military or firefighting services, or the aircraft used by Orbis.
There are 255 active of 561 delivered. Excluding 6 in military or other service, this leaves 178 in cargo service and 71 in passenger service.
There are about 821 767 aircraft in service of 1062 delivered. About 165 are cargo versions, and excluding 16 in military or other service, this leaves 640 in passenger service.
There are 105 active of 255 delivered. Excluding 17 in military or other service, this leaves 14 in cargo service and 74 in passenger service.
There are 153 aircraft in service of 200 delivered. Only 3 of these are passenger aircraft. These are operated by KLM, and the last scheduled flight is October 25, 2014.
There are 1069 active of 1112 delivered. Excluding 22 in military or other service, this leaves 27 in cargo service and 1020 in passenger service.
There are 289 active of 377 delivered. Excluding 11 in military or other service, this leaves 278 aircraft in passenger service. If there are any versions used only for cargo, there aren't many.
There are about 1216 777 aircraft in service of 1230 delivered. About 134 are cargo versions, leaving about 1082 in passenger service.
There are 139 delivered. There are no cargo variants and none out of service.
There are 183 delivered. There are no cargo variants and none out of service.
There are several ways to estimate the answer. One would look at the number of delivered airplanes per year, and the other would start from the total number of passengers per year. See here for data for both methods.
Airbus and Boeing together deliver almost 1000 airplanes per year. If we set a lower limit of 150 seats for what counts as a large commercial aircraft, 1000 planes would already be a good estimate for global production. If we now assume that an individual aircraft will fly for 30 years and factor in the growth in delivered airplanes per year, the worldwide fleet would be around 12.000 aircraft. There are a few airplanes which start their life as cargo aircraft, but most are converted from old passenger aircraft. For now, I would assume that all newly built aircraft will go into passenger service.
How many airplanes would be needed to ferry 3 billion people per year? Short-range aircraft can fly 5 times a day when well managed, and long-range aircraft still manage more than one flight per day. Since the majority of flights is short range, the average number of flights could be 4, with 230 seats and 75% load factor on average. This comes out to 3.000.000.000 / 4 / 172.5 / 365 = 11.912 planes.
Looks like 12.000 is a reasonable estimate for the worldwide total of large commercial passenger aircraft.
Using ICAOS number of 16 billion freight ton-kilometers of air cargo volume worldwide, and guessing an average load of 30 tons over 2000 km, it would need 266.667 flights to ferry this volume. If a freighter flies twice a day, the total number of freighters could be just 365 aircraft. This would be just 3% of the number of passenger aircraft.
Edit: The first estimate leaves out all aircraft smaller than 150 seats, which should be included in the ferrying of those 3 billion passengers. I think my utilization estimate is too high; a better number would be 2 flights per day. This moves the total number of commercial passenger aircraft to 24.000, but includes small ones which a correct answer to your question should not include.
So my money is still on 12.000 for aircraft bigger than 150 seats.
Also, cargo aircraft should be more numerous; again my utilization is on the high side, I guess. But I would be surprised if there are more than maybe 800 big cargo jets in operation worldwide.