Neither Bitcoin or gold is rare, anyone who wishes to can purchase a Bitcoin or a fraction of it, same goes for gold. Even if it were rare, that is not where its value comes from.
Of course it is. Something that is widely demanded but scarce in supply (like gold) has high value. Something that is widely demanded but plentiful (like iron) is cheap. The reason for that is scarcity.
People call Bitcoin digital gold, because they want to make an argument that it's a hedge against inflation. The problem is, Bitcoin is not showing any good correlation with inflation or other macroeconomic indicators, it's completely unpredictable. The main point of a hedge is to act predictably under some circumstances, and Bitcoin still can't do that.
I'm not sure about all that. Is it actually true that Bitcoin doesn't negatively correlate to inflation? Either way, I don't see that as a requirement for people to treat BTC as an inflation hedge. I wouldn't call it completely unpredictable either, any more than stocks or gold anyway. As risk assets they are all correlated, particularly on higher time frames.