Honestly, Spike shares the exact same view on the matter as I do. My friends actively mock furries and all that but are still friends with furries. My best buddy Troi was not just a furry, but he was also a Pokemon furry. He would always be acting "Buizelly" as he called it, considering his fursona was the Pokemon Buizel and that Buizel was named Buster. My friend Trevor is an active furry still, and his is a fox named Sparky. Likewise, there was Chad, Geoffry, Chance and many others who were open furries, wearing tails and all that around school. (No full suits because masks are not allowed.) But despite this, my "furry hating" friends are never serious and appreciated our company as sincerely as non-furries, as did all the other people I had befriended, whether they knew initially or not.
Yes, they got their jeers and odd looks, their fair share of laughs and all that. But it will honestly fade if you simply treat it like a normal thing and being confident about it. It's a matter of not showing fear and keeping your head held high, they will see you do not feel it is something you should be ashamed of, and they will back off. And if anyone bullies you for it, well, keep in mind that the staff members are there to help you, as that is part of their job.
Overall, many people are much kinder than is let on or perceived, this is an era of changing minds. Don't show shame in the things you enjoy, because it's your life and nobody else's, and that also puts it in their head that your interest is indeed a "shameful" thing, which they take as a green light to mock it. Being a furry isn't shameful, it's harmless. The sexual things you do are for you to know, however, and nobody has to know if you don't want them to.
I also have to say that your sister should be ashamed. As an older brother of three siblings, I make sure that they all feel comfortable talking to me about anything and I never make fun of them for their different views or interests.