Basically, a sine wave is a special case of a sinusoidal wave with zero phase. Now talking about frequency, it is the number of cycles that happen in one second. You have got a wrong conception that frequencies are present for only sine wave. But any periodic waveform will definitely have a frequency let it be a triangular or sawtooth or square wave etc. And for non periodic signals, there will be a maximum frequency content preset provided the signal is a mixture of many frequencies.
Regarding the bandwidth,let us take the human ear. Human's are capable of perceiving sounds in the range of 20Hz to 20,000Hz. The bandwidth is defined as the range of frequencies in which a system would operate. In other words, it is the difference between the highest and the lowest frequency components in a signal. The bandwidth of the human ear is 20,000 - 20 = 19,980 HZ.
Another example is the voice signal. Most of the information spoken by humans are in the frequency range between 300 Hz to 3,400 Hz. Hence the bandwidth of the speech signal is 3,100 Hz. Generally speaking, Bandwidth is the amount of space occupied by a signal in the frequency spectrum.
And for the first question, yes it is possible to have many frequencies in a single wave. That's what Fourier series is all about. According to it, any periodic signal can be represented by the sum of 'N' number of sinusoidal waves of same or varying amplitudes. the sine waves have a definite frequency called as the fundamental frequency and a number of harmonic frequencies which are the integral multiples of this fundamental frequency. In case of a non periodic signal, a Fourier Transform is used which assumes the whole signal as one cycle of a periodic signal and does the same procedure.
Hope u are able to understand me !!!