Your sentence uses the verb phrase "to have to". This is simply another way of saying "must". You could reword the sentence as: "...everything must end sometime" and not really change the meaning at all. Both of the "happy" sentences are grammatically correct, but you never say/hear "I feel happiness" in the context you are likely describing. Spoken American English prefers you use the adverb "happy" to modify the verb "to feel", rather than use "to feel" with the noun "happiness". In short, stick with "I feel happy."

I'd=I would I would be miserable(if something ended...a relationship?) but everything has to end sometime, so I would be fine after a while. This person is talking about the end of something. He/she would be very sad at first, but would be fine later. It's far more common to use a the adjective form "happy" with the verb "feel". "Happiness" isn't incorrect, but it sounds strange.