How do you derive steady-state conditions from the open-state differential equation of ion channels?
These are really math problems (algebra and basic calculus) rather than biology problems.

Oinf comes from solving algebraically the first equation you posted for steady state: i.e., when dO/dt = 0; just solve for O.

The time constant comes because this is a first-order exponential decay, which you get when you solve an equation (calculate an integral) of the form dO/dT = -lambda * O. The solution to that equation is an exponential decay of the form O(t) = O(0) * exp(-lambda*t)

tau is defined as 1/lambda and gives you the decay rate(relaxation time constant).