Wow, the answers to your questions could fill books. Just a start...
1. SMAW, FCAW, SAW, etc.
2. One standard is AWS D1.5 Bridge Welding Code. Check the AWS site for information on how to order.
3. VT, UT, MT, RT, etc. plus destructive tests like tensile, Charpy v-notch, hardness tests, etc. when qualifying procedures, etc.
In Canada we use similar welding and testing processes as mentioned in the previous reply. Our welding and design standards are CSA W47.1, CSA W59, CSA S6 (for Highway Bridges).
You may also want to try the United States Federal Highway Administration's website for more bridge related info. There website is http://www.fhwa.dot.gov
-Sean