Is this common or was I just unlucky?
Neither. The primary problem, presumptively based on your description, was a combination of inexperience - overall, and with the equipment you were using - and lack of proper pre-dive preparation. Luck had nothing to do with it. While that may sound harsh, it is not intended to be at all - merely a statement of reality. You have had a very valuable learning opportunity, from which you returned safely. That is worth a lot!
The situation seems to involve 4 elements: 1) a relatively new / inexperienced diver, 2) using unfamiliar equipment, 3) which may not have been properly adjusted, and 4) failure to follow optimal procedures (pre-dive checks). And, just so you don't feel that I am picking on you, remember that most of us have been through similar 'growth opportunities' as well. We all have to start somewhere.
Based on your description, it is likely that 1) you were somewhat over-weighted and 2) your weight belt was either not properly secured / tightened, or was possibly somewhat worn and (particularly if you were using uncoated lead weights) allowed the weights to slide around on the belt. But, even if weights shift, the properly weighted diver will usually be able to compensate for the effects on trim. The experienced diver will also be able to make adjustments during the dive. One of the reasons that 'weight belt remove and replace underwater' is taught in most OW courses is to provide the diver with various skills to address these kinds of issues.
MrSam:
my buddy and I are in the water ready to descend. I noticed I was wobbling a bit but I figured it wasn't a big deal...big mistake.
Yes, a mistake. To paraphrase one wise SBer (with whom TSandM is somewhat familiar), 'Nothing gets better underwater'. So, while frustrating, this was a valuable lesson to learn - a proper pre-dive check will include checking your weight system - total amount and distribution - and help minimize such problems.
MrSam:
We also found an issue with the BC. We realized that about 75% of the air had gone to one side of the BC and 25% to the other. There were some "cords" of some sort holding the bladder in place, and apparently one side got tangled up and restricted air flow.
This is really not an issue with the BCD, as much as failure to check the equipment beforehand, and possibly the use of an excessive amount of air to compensate for an excessive amount of weight. Again, a proper pre-dive check should identify such problems. The 'tangled' cords were apparently obvious enough to be identified BETWEEN dives. Therefore, they should have been obvious enough to be identified BEFORE the dives.
MrSam:
Is this a common issue with back inflate BCs?
Not at all. Despite periodic negative comments regarding bungees, they are not inherently 'bad'. If improperly used, just as with any equipment, they can potentially be problematic. And, a number of divers have elected to avoid them altogether. I own and dive a number of wings - bungeed and not bungeed, horseshoe and donut, small and large. While I have definite preferences - for example, I like a small, unbungeed donut wing for single cylinder recreational diving - I don't have a real problem with any of them. I also don't have a problem with a jacket BCD (unless I am over-weighted and the weight is improperly distributed or secured).
MrSam:
What is the main complaint that people have with back inflation BCs? Is it just the "face plant" at the surface thing?
This is probably the complaint heard most often. But, it more an 'operator error' issue than any design issue. In the situation where 1) a back-inflate BCD is used by an over-weighted diver, and 2) the BCD is weight-integrated and 3) the weights are positioned toward the front of the diver (where too many weight-integrated systems improperly place them, anyway), the diver may feel that they are being pulled down slightly toward the front at the surface when they inflate the BCD to allow their head to be well above the water - the center of lift (behind the diver's head at the surface), and the center of gravity (in front of the diver at the surface), are inexorably moving toward vertical alignment. This has to do with simple physics principles as well as BCD design. To compensate for this feeling, the diver (understandably) adds even more air to the BCD, which then causes the bladder to rise even more behind the diver's head, exaccerbating the feeling of being pulled face-down.
As NetDoc says, dive with what you are comfortable. There are a lot of jacket BCD divers out there. But, as others have also said, don't necessarily limit yourself this early in your diving, by over-reaction to a problem that has more to do with your experience than it does with your equipment.
chrpai:
One bad experience, one good experience and he's running back to what he knows. Oh well. . . . A balanced rig with a minimally sized donut wing is all I need for single tank diving.
Well-stated.