no issues for 1,2. CBP usually asks where you are going. Just respond them that 'you are going to work for <your-employer-name>' every time and don't volunteer any other information unless asked. As always, keep your LCA and employment letter stating your work location handy. 99% of the time they don't look in to them, but they just make you feel better and confident.
For nexus, you definitely need valid visa at the time of applying as the nexus application asks for visa number under that documents that you have to prove that you can enter US. Anyways people are not getting Nexus approved these days as the timeline is 12-24 months. So even if you have valid visa and apply, you are looking at 1-2 years for any update on your application.