Cleanliness...it doesn't take much grit in these carbs to make them malfunction in the little ways you describe.
Take a look at the little emulsifier holes in the pilot jets and the mainjet "holder", make sure they are CLEAN. I use a sharp, pointy thing to make sure, and sometimes drill them out a couple thousandths (mainjet) if they are the holes of 0.033" size. With ethanol and unleaded gas, they should be at least .035" size in the larger 8 holes of the mainjet's emulsifier, which will mix better.
The backfire: this suggests the carbs are not balanced. If you don't have a set of vacuum gages around, try first to figure out which one is popping, then adjust that one's slide upward about 1/8 turn (of the 17mm nut) at a time until it stops popping on decel. This will get you pretty close.
In the 657A carbs, the ones made 1971-1972 January usually came with #110 Keihin jets. After that, they were #105 jets. The aftermarket jets we are getting in carb kits like Keyster or K&L are often size-marked "110" but are really closer to #103 or even #100 in some cases I have seen recently. You can get real Keihin jets at Honda or Jets-R-Us online.
Also: if the carbs were rebuilt with non-Honda float valves, the springs in the little float valves will be very stiff. This can result in low bowls levels when running, so I usually set these floats to 25mm instead, to keep the bowls a little "wetter" during acceleration.
If you have a strobe timing light: make sure the timing is on the marks, and that it does not "jitter" back and forth, especially on the 2-3 points side. If it does, the points' advancer shaft has been bent (caused by turning the big nut CCW with sparklugs installed), and needs to be straightened to within 0.002". I show a picture of how to do this in my book, if you have that?
