Laffer curve in one period economy
Disclaimer: I still have to educate myself on utility theory (same as I disclosed here).


  Would production possible without the government: t= 0?


It cannot be determined with the information given. The problem's definition of $T$ is unclear or incomplete, as the only component of $T$ mentioned in the problem is the physical capital tax rate $t$.

Assuming that the only source of government revenue is the physical capital tax, then no, production would not be possible if $t=0$ because --as you point out--, the latter implies that $T$, $H$, and therefore $Y$ are zero. 

But that prompts the question: Does the definition of production $Y$ even make sense if  = ? Obviously not: Production should still be possible because private physical capital $K$ is not affected by the condition $t=0$.


  Do households have incentives to rent the physical capital to the firm
  if t= 1?


Yes, they may have an indirect incentive even if $t=1$: The real wage $w$. If nobody rented physical capital, there would be no demand for labor. Consequently, no household would earn income from labor even if a household's utility function indicates a preference for work over leisure.

I gather that the problem is a textbook exercise, and/or that not all the assumptions are reflected in the inquiry. Thus, a portion of my answer possibly goes beyond the scope of that chapter or section.