expanding universe or shrinking space(-time)
1. If you ask me BB model is on the right track, but still has a potential to throw some surprises, probably as soon as James Webb telescope starts beaming pictures back to Earth.
2. You are probably referring to cosmological redshift, which describes redshift of distant objects as "stretching" of light wave during its journey through expanding space.
3. Hubble law is straight forward observational law, it really does not need much proving.