I am working with variable capacity utilization like in Jaimovich and Rebelo(2009). In my model, the technologies are random walk with drift.

Do I need to transform/normalize variable capacity utilization equation to make it stationary?

As far I know, capacity utilization is bounded between [0,1] like labor hour. In my model, I am not normalizing labor hour; should I do the same for capacity utilization?

No, with the typical function forms (see e.g. dx.doi.org/10.1016/j.jedc.2013.06.011, equation (9)), there is no reason to detrend capacity utilization, because it is stationary.

Yes, the Lagrange multipliers also have trends. Take the Lagrange multiplier on the budget constraint. It is equal to marginal utility of wealth. With CRRA you have

Consumption will grow with the rate of output along the BGP. The Lagrange multiplier therefore grows with the rate of output raised to the power of -sigma.