If I take an option that gives me a spell (For example, Seer elf or Arcane Sense) and then I take a sorcerer bloodline that gives me the same spell (Say, Imperial bloodline which gives you detect magic as a cantrip), do I just lose out and not get the extra spell, or do I replace it? I have not found anything about spells, I know skills allow you to select another skill. Do spells work the same way, or is there a different rule?
Neither. In the examples you provided you would have the spell as both an Arcane Innate spell and an Arcane Sorceror spell. Because those are both Arcane and Charisma based, it would be redundant, but technically you would have 2 versions of the spell.
Yes, you would have both.
Innate spells can be nice because they don't necessarily have visible casting manifestations or spell components like traditional spells do (though this has been debated).