No, it does not have to be periodic. "Perfect reconstruction" only requires that the original signal sampled by the ADC sampled was bandwidth-limited.
Nope. The signal in this image is bandwidth limited (5Hz signal, 11Hz sample rate) but sin(x)/x won't reconstruct it unless the filter is infinitely wide and the signal is periodic (which can't happen in practice).
man what are you smoking? please focus on gf's word here
"realizable real-world reconstruction/interpolation filter"... true Sinc by theory cant be implemented in real world, so professors have to find a way to make it "realizable" or "practical", similar to Fourier Transform, without DFT or FFT, real Fourier Transform will not be "realizable" because real FT is infinitely long. go read some literatures on how to make those "realizable", i cant comprehend them myself but at least i got some idea. the FFT you keep bragging about, is one crippled version of FT that brings some complications with it, ymmv.