They ask if I have made an application to the Home Office to remain in the UK in the past 10 years.
I answered no because I think a fiancé visa application is different from asking for leave to remain, correct?
Yes, that is correct.
Leave to remain can only be applied for from inside the UK (i.e. you are already there and are applying to stay (to remain) in the UK).
They ask if I have ever been denied entry to the UK.
I answered yes and put in the reference number.
But you haven't been denied entry to the UK - you were denied entry to Ireland, which is not part of the UK.
So, I would answer 'no'.
Then they ask "have I been deported, removed, or otherwise required to leave any country, including the UK".
Should I answer "yes" since I was denied entry to Ireland on the basis that they thought I was trying to travel to the UK? Does "denied entry" fall under the category of "otherwise required to leave"?
No, this doesn't apply to you.
Being refused entry is completely different from being deported or removed - you didn't even enter Ireland, so you couldn't have been deported or removed from Ireland.
It is asking if you have ever been living in any other country illegally and been arrested and deported or forced to leave by the government.
For my marital status, there are options for both "single" or "fiancé/proposed civil partner". If you select fiancé, it asks for their details and current address.
Should I write “fiancé" since we applied for the fiancé visa? Would they know about the fiancé visa application? Would either answer “single” or “fiancé” count against me?
You need to be honest.
If you are engaged, then you are not single, so it would be lying to say you are single.
I'm assuming you have withdrawn your fiance visa application? They may have a reference of it in their system, but I'm not sure.
You can always include a cover letter in your application package explaining that you mistakenly applied for a fiance visa, but your fiance is an EEA citizen and didn't qualify to sponsor you, so you withdrew the application.