Leonardo, Good day sir,
You have asked a multi tiered question. Lets start with the material ID:
Remember, you are dealing with more than just D1.1 or 1.5 on a job. You must also consider AISC Construction Manual (I am referencing the 13th Edition) which contains the Bolting (which doesn't concern us at this point), the Specification for Structural Steel Buildings (which somewhat concerns us) and also the Code of Standard practice (which really concerns us). Look at the Code of Standard Practice (COSP) at the back of the manual at Section 6, especially 6.1.1 and 6.1.2. They are going to direct that material should have material id visible through the fabrication until assymbly. When in connection with a shop that would indicate that as it comes off the truck it should have some id (at the least a Heat #) from the supplier and that id follows it through the detail prep lines of saws, torches, whatever to make all your small components. This will cross over to many codes from D1.1, 1.8, 1.5 and others because it is the COSP and is to be considered standard practice in all work.
The TPI needs to be able to view this as much as possible to document that the correct materials were used on the job. Some small parts should at least be visible until fit up at which point it will be difficult to keep the Ht# (which is most often used for traceability) visible and after welding even more so.
You will also find some references to this marking/id of parts in your applicable codes especially if it is a Seismic job (D1.8 and/or AISC Seismic Design Manual), Bridge job, or Pressure Vessel job.
But mainly, you are going to find it in many larger company QC manuals of operation. It is their standard practice to make sure that it doesn't get missed on jobs that do require it. Every part gets marked with job#, heat # and a couple other numbers depending upon the company to be able to track material. Yes, EVERY PART.
Secondly, the welder. Now once again you are going to have to check multiple codes. It is in D1.8 Seismic Supplement Clause 6.13. It is a good practice for keeping track of welders and possible additional training that may be required. I believe you will also find it in the Bridge Code (D1.5) and your Pressure Vessel Codes.
Again, it is good practice just for records of UT and RT acceptance percentages and the need for additional training. BUT, it is required for some codes. When working to the multitude of codes that you appear to work to it is good practice in order to keep the welders in the habit. Once again, it may be one of those customer requirements and/or Fabrication QC items that isn't always a code requirement.
There are many things to consider with each of your questions. If a company just establishes both as Standard Practice then they can't go wrong.
Have a Great Day, Brent