Your situation (1) happens with the equivalent of a research assistantship funded by some company; this is common in fields where companies want to hire PhD students to do projects for them, and not common elsewhere. Otherwise you're just a person taking your own salary and spending it on a graduate program; that might happen especially for students who have already exceeded funding limits but still want to graduate, but it's a really bad idea to try to start out that way. Individual programs might allow it anyways.
For (2) you'll have to check for any specific funding stream, but generally I would expect you will not find policies that say "you can't work", but will see policies that state you need to be fully dedicated to research or something along those lines, and that it's up to your institution to make sure you're following those guidelines (in the US, the government generally tasks academic institutions with the administrative burden of making sure their money is spent well). I'd expect any reasonable university policy to say that holding a whole additional full-time job would not permit students to direct their full effort towards their studies.
For example, from this Q&A: https://www.nsf.gov/pubs/2013/nsf13084/nsf13084.pdf
I am on Tenure; may I participate in paid/unpaid activities?
Fellows are expected to devote full time to advanced scientific study or work during tenure.
However, because it is generally accepted that teaching or similar activity constitutes a valuable
part of the education and training of many graduate students, a Fellow may choose to undertake a
reasonable amount of such teaching or similar activity, without NSF approval. It is expected that
furtherance of the Fellow's educational objectives and the gain of substantive teaching or other
experience, not service to the institution as such, will govern such activities.
Compensation for such activities is determined by the GRFP institution and is based on the institution's general employment policies. Fellows are required to check with their GRFP institution about specific policies pertaining to GRFP fellowship and paid activities.
I am on Tenure; may I work an outside job?
Outside employment is not governed by the NSF. Fellows should check with their CO about specific institution policies.
("on Tenure" here means "actively being paid by the fellowship")
It's not normal for people to do a full-time job and separately do full-time PhD work; it's impossible to do both at a high level. Don't dismiss these concerns about stress/work/life balance just because you think you can do it: the warnings exist because of all the people who thought they could but couldn't.