DSP: Designing a FIR Lowpass Filter / Calculating Coefficients?

If the sinc is centered in the middle of the buffer, is the convolution still performed with both initially starting at zero? As in the first value is

y[0] = x[0]*fir[0] + x[1] * fir[1] + … + x[buffSize-1] * fir[buffSize-1]
presuming x and fir are the same size. next value being
y[1] = x[1]*fir[0] + x[2] * fir[1] + … + x[buffsize - 1] * fir[buffsize-2]
etc.

Or does the convolution perform from a different starting point? I guess I’m confused in that I’d figured centering the Sinc in the fir buffer would get you a different output.

Instead of worrying about how to do the time-domain convolution I’ve also been trying to figure out how to do a frequency multiplication with JUCE’s FFT class, but I’m stumped there too- Multiply results of FFT class? (convolution)