Swinging the hammers myself to save on labor, where does that stop being allowed?

Owner-builder permitting varies by state and often by city within a state, so both things you were told are probably true somewhere. Common shapes: some jurisdictions let any property owner pull permits on a building they own, some restrict owner-builder permits to owner-occupied single family, and some allow it for any owner but require a licensed contractor for specific trades regardless. Electrical and plumbing are the two that get carved out most often. Your local building department will tell you the rule for your address, and if the answer isn't clear you want a licensed professional to read it rather than a forum.

The appraisal question is separate and the answer is that the appraiser mostly can't tell and mostly doesn't care. Value comes from what's there, not who installed it. Your labor doesn't show up as an extra credit, so if you save 12k in labor, that 12k stays in your pocket as reduced spend and doesn't appear as extra value.

What can affect the refinance is permits. If the county has an open permit on the property, or if a lender or appraiser spots work that clearly required a permit and no record exists, that can hold up a closing until it's resolved. Closing out permits after the fact is slow and sometimes means opening a wall you just finished.

One more thing that catches people. If you're funding the rehab with a construction or hard money draw, many of those lenders release draws against licensed contractor invoices and won't reimburse your own labor at all. Ask before you plan the budget around self-performing.