Is there a way to mix a iron golem farm and a villager (breeding) farm?
You don't need to constantly produce villagers in order to have an effective iron golem farm. Two basic conditions need to be in place for an iron golem to spawn:
Villagers: Iron golems spawn if there are at least 16 villagers nearby. For every 15 total villagers, one iron golem may spawn (16 = 1 golem, 32 = 2 golems, 48 = 3 golems, etc.)
Houses: It takes at least 21 houses to spawn an iron golem. A "house" in Minecraft has a complicated definition; here's the simplified version on the Minecraft Forum:
A block counts as being part of the "roof" when it is within 5 blocks of the door (in the two directions the door faces), its topside is exposed to sunlight during the day, and its bottom side is higher than the door's bottom side. Notice that a door is not a "house" without a roof, or with equal amount of roof on both sides.
That description is slightly misleading as the algorithm is more complicated. For more information, I recommend reading Marfagames' thread on the Minecraft Forum.
Generally speaking, with both those conditions satisfied, iron golems will spawn. I encourage you to read up on the specifics if you wish to design your own farm. Lower down I also recommend a specific design.
Farming Tips
Having more than a greater number of villagers will not affect the rate at which iron golems spawn. The number of villagers only determines the iron golem cap.
If you drop the iron golems at least 5 blocks down from your farm, they will no longer contribute toward the iron golem cap, and more will continue to spawn.
A house only becomes part of a village if there is a villager within 16 blocks horizontally and 4 blocks vertically.
More Resources
JL2579 over on YouTube has made several videos on iron golem farming, and has designed the most effective golem farm I've ever encountered.
Micromancero came up with a farm design that inspirqed JL2579 (on YouTube).
Docm77 has done a lot of work with JL2579's farm on YouTube.
Note: I would love to post more links, but I don't have enough reputation. JL2579, Micromancero and Docm77's videos can be found easily via YouTube search.