Why does Madara appear unhurt despite being covered in wounds? I'm not sure if I have missed anything in the past, but Madara seems unphased by every attack.

Sasuke stabs a sword through his arm, but this isn't a problem.

All the tailed beasts stomp on Madara with their tails, and Madara doesn't even evade. Instead, he tanks the attack like it is no big deal.

What's going on here? Hashirama's senjutsu should only do so much, but it seems like Madara is simply invulnerable, aside from a little bleeding. Why isn't Madara getting hurt at all?

Is this what we call a plot shield?
From chapter 659,

Madara mentions that due to the healing powers of Hashirama's he is able to recover must faster and is taking close to no damage.
So it is not just just Hashirama's senjutsu, but also his healing powers that have made Madara virtually invicible.
Also from the same chapter:

It is not that he is not being damaged, but that his healing is so fast that all damage seems to have no effect at all.