Derivation of time constant for single source, single capacitor RC circuit
1 and 2 are true because you can derive it from first principles using differential calculus. See:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capdis.html

and

https://www.york.cuny.edu/academics/departments/earth-and-physical-sciences/physics-lab-manuals/physics-ii/time-constant-of-an-rc-circuit

The relationship only holds for circuits that can be simplified to an R in series with a C. For other arrangements the equations would be different (tho likely take on a similar form).