The 300 ppi is a norm derived from colour printing with offset, where the images had to be transformed to grids of separate points. Iirc, those grids were closer to 150 dpi, which corresponds to 300 ppi (1 coloured point + 1 space). Also, the result was expected to be viewed from normal reading distance (~30cm), where the human eye just can't distinguish the individual dots.

That means you should aim for 300ppi for photo prints looked at from ~30cm on smooth paper, and progressively lower when the viewing distance increases. As soon as there is any texture in the paper, you can go to somewhat lower ppi (Some say you can go as low as half the recommended ppi on canvas, which has a lot of surface texture).

If you increase the resolution in photoshop to make larger prints, quality will go down (details will start blurring), as you just don't have the information there to go to the higher pixel counts. But if you have to do that, some extra sharpening after the enlargement can help.

I understand you already have the files you have to use, but starting from the originals, you can get sharp prints larger than the original by e.g. scanning at a higher resolution. But this will also enlarge any defaults present in the originals.

And of course, to judge the quality of the resulting prints, you can just print a standard 4"×6" from an interesting part of your final image file.