Sine of frequency 0 contains sines of all frequencies at once in it
You are confusing time domain and frequency domain. A constant time domain function (a sine with frequency 0, if you like) corresponds to a delta impulse at the origin (i.e., frequency zero!) in the frequency domain. A delta impulse at frequency zero is zero for all other frequencies $\omega\neq 0$. Consequently, a constant in the time domain is not a combination of all frequencies. What you were probably thinking of is a delta impulse in the time domain, which corresponds to a constant in the frequency domain (i.e., it contains all frequencies).

So, summarizing, a constant in one domain corresponds to a delta impulse in the other domain. A constant in the time domain does not contain any frequencies other than 0 (delta at zero in the frequency domain). A constant in the frequency domain (i.e. a combination of all frequencies) corresponds to a delta impulse in the time domain.