In a nutshell :
When submitting a paper, make sure the acknowledgement reflects
This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
It is the same contract — and acknowledgements– for everyone at the lab. You can mention “partially supported by” or equivalent when needed. The statement might vary based on which specific office funded the research, but the contract number should always be acknowledged.
You can check the copyright status of your publications on SHERPA/RoMEO. It is good practice to make sure that all your publications can be found on the internet (either through Google scholar, PubMED and others.) You can also use your own website to make your publications available (the worst things that can happen is a take-down notice, if the copyright situation is tricky.)
Regarding intellectual property – please submit a record of invention to the Berkeley Lab Innovation and Partnerships Office (ipo.lbl.gov) for new processes, devices, or compositions of matter described in a paper when you have completed a draft of the paper. The record of invention form can be found here. You can find more information here.