Understanding Free Cash Flow more clearly

I've always learned that:
unlevered FCF = EBIT(1-t) + DA - Capex - change in (operating) NWC
levered FCF = net income + DA - capex - change in (operating) NWC
unlevered FCF - tax effected interest = levered FCF
tax effected interest = I*(1-t)

a few questions.
1) how come sometimes I've heard it phrased as change in NWC? shouldn't we always exclude cash and short term debt and use the "operating" NWC?
2) likewise, when someone asks about net working capital, it's current assets - current liabilities, but I assume they really mean (operating)net working capital especially in the context of building cash flows
3) when would we look at net working capital (including cash/short term debt) aside from examining liquidity?
4) For an LBO when we are looking for CADR (cash available for debt repayment) doesn't it make sense to account for all changes in cash? So should we also include changes in cash from financing and cash from operating changes in other long term assets and other long term liabilities? In other words, is it better to claim that CADR = LFCF (and think our initial LFCF definition incorrect and therefore expand it) or CADR = LFCF (per our initial definition ) + adjustments (financing, LT assets/liability changes)?
5) Following the above point, then does the calculation of UFCF and LFCF for TEV and Equity value DCF makes sense? If we're trying to value the business/company as a whole can't we argue that financing is also part of its cash flows?

Thanks.