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1. Is this sentence correct? Why not delete the preposition "on" and write as : Which day was your last anniversary?

On which day was your last anniversary?

2. I thought the uses of the verbs "do" and "are" are very different and leave very little room for confusion but are these sentences valid? Their meaning seems to be slightly different though.

Do you feel like a king?

Are you feeling like a king?
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2. Are you feeling like a king?

I don't think this is right, feel is a non-progressive verb:
http://www.english-zone.com/verbs/prgverbchrt.html
Hi guys,

1. Is this sentence correct? Why not delete the preposition "on" and write as : Which day was your last anniversary?

On which day was your last anniversary? Either is OK. It depends on whether you are thinking of the anniversary as an event or as a date.

2. I thought the uses of the verbs "do" and "are" are very different and leave very little room for confusion but are these sentences valid? Their meaning seems to be slightly different though.

Do you feel like a king? Are you feeling like a king?

Obviously, you need a context for this 'king' stuff. Your question really seems to be about Simple Present versus Present Continuous. Let's look at simpler examples.

I feel sick. I am feeling sick.

These both sound acceptable to me, and for these examples there is very little difference in meaning. Possibly #2 sounds just a little more temporary, a little less definite.

Best wishes, Clive
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There's always the possibiliity of asking What day did your last anniversary fall on?
CJ
With all due respect, I know the collocated words "fall on" is perfectly acceptable in the context you had it on/in?? But to use it to refer to the date or day of a person's anniversary gives a rather worn-out impression on the occasion marked as the special day worthy of your remembrance.
Thank you.

Could I make the questions out the simple examples you gave like this?

I feel sick -- Do you feel sick?

I am feeling sick -- Are you feeling sick?

And both would convey the approximately (not that I am measuring anything) the same meaning?
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Both of those are OK.
With all due respect, I know the collocated words "fall on" is perfectly acceptable in the context you had it on/in?? But to use it to refer to the date or day of a person's anniversary gives a rather worn-out impression on the occasion marked as the special day worthy of your remembrance.