Electronic – Why is there a “ – 1” in the pn-junction current equation

Separately processed pieces of semiconductor mechanically joined won't work well as a pn junction because the facing surfaces need to be perfectly joined (as the crystalline structure level), and without and intermediate impurities (oxidation of the surface). In theory it would work, if you could align the crystals perfectly, but in practice it can't be done.

Generally a pn junction is formed by taking a bulk doped semiconductor (e.g. n-type), and diffusing p-type impurity at the surface. This creates a pn junction beneath the surface of the semiconductor. The p portion is actually counterdoped - it contains n impurities, but more p impurities, giving a new p-type characteristic.

How can a charge neutral substance have a potential?

To say that a PN junction has built-in potential isn't to say that the PN junction has a potential relative to ground or infinity etc.

How can there be an inherent potential in a doped semi-conductor if it is charge neutral?

Charge has been separated within the PN junction and, thus, there is an electric field across the depletion region and an associated potential difference. A charged capacitor is neutral but there is a potential difference (voltage) across the dielectric. There are some similarities but...

However, I don't understand why we can't measure the voltage drop across the PN block with a voltmeter.

As explained, for example, here, the built-in potential is not readily measured with, e.g., a voltmeter. In other words, this question has been asked here several times (which means you are not the only one perplexed by this - most are at first) and there are good answers already available.