So I am aware that when you round to n decimal places accuracy, the maximum error is $~0.5 × 10^n~$
But if I use the rounded result, and use multiply it by another un-rounded number and round the result again.
Multiple times, does the maximum error of $~0.5 × 10^n~$ still hold?
The error at each stage is limited to $0.5 \times 10^{-n}$ but how that propagates through a calculation depends on the calculation. It also depends on whether you are talking about relative error and rounding to $n$ significant figures or absolute error and are rounding to $n$ places beyond the decimal point. For relative error, multiplying by something does not change the relative error, but you commit another one when you round again and your relative error can now be $10^{-n}$. Using absolute error if you round to $n$ places then multiply by something of order $1000$ the error can be about $0.5 \times 10^{-n+3}$ before the second rounding.