I also used the sugar solution in the bucket method my first two bottling efforts and all mine came out evenly carbonated.. It's what I would recommend. I also used the priming sugar (Corn..?) that came with my kit, not sure if that matters. Definitely made it easier to bottle. I can't imagine the pain and mess it would be putting a spoon of sugar in each bottle before bottling.
Also I am stickler for sanitation.. so no infections in any so far that I have opened up.
What I did with my bottles was this :
1) Soaked all my bottles in hot Oxyclean bath for a half hour and scrub the outside (Oxyclean strips the glue and labels off with little effort). Make sure the bottles are all filled and sink to the bottom.
2) Rinsed them off and then use a bottle washer (with hot water) to rinse out anything inside the bottles.
3) Then I go to the dishwasher and put them in the dishwasher (Without soap) and run the dishwasher on sanitize. So even if you miss something in the previous wash, they will be sanitized.
4) Removed from the dishwasher while still hot and put into their 6 pack containers to dry and cool off. Which will be fast as they will be 140 degrees or so.
5) While they dry I prep for bottling and then bottle. By this time the bottles are cool enough to handle, but still in the 80 degree range.
The heat transferred is to the beer is minimal. My thought is if the bottles are little warm, then the colder beer will warm up (at least around the edges) a bit to kick off the yeast to start making some CO2. Especially for yeast that at the bottom, the warmest part of the bottle.
The filled bottles by the time they were ready for capping were room temp.
My first batch have a nice even carbonation and head 2 weeks after bottling. So it must have worked. I would say with all the mistakes I made in the brewing aspect of my first batch, the bottling wasn't one of them.