Happiness is more usually used.

Yes, it's old-fashioned. I saw the word "felicity" the thing that popped into my mind was the line from Shakespeare's Hamlet, "absent thee from felicity awhile," written in 1600. Shakespeare's English is so old-fashioned that most native English speakers--if they are honest anyway!--would admit to having trouble understanding it. I'm not sure I have ever seen "felicity" used anywhere else. "Felicitous" is a little more common, but it is a somewhat "literary" word. It can mean "very nicely phrased" or "very appropriate."

'Felicity' is not a normal everyday word at all in modern English. You might see it in a poem or text from several hundred years ago, but nobody would use it now. In fact I would guess that a large proportion of native speakers wouldn't even know what the word means. It is is girl's name, though.