Workflow for trimming, encoding, and publishing videos to the web
If you are willing to use Adobe products, Premiere supports direct integration with Adobe Media Encoder and can be set with presets that include multiple output formats and sizes.  Each encoding runs off the actual main project, thus you remove the extra generation of loss.

Alternately, using a lossless quality intermediate file will prevent issues as no additional quality will be lost going from the source material to the intermediate output.

A big pipe is the only answer to the second problem.  Encoding to the final format locally will save on bandwidth, but you have to transfer the file at some point and final output is the smallest amount of data possible unless you are making an awfully large number of different sizes of each video.