These are genuinely great questions — the kind that most students don't think to ask because they're busy writing things down. Let me give you the context you missed.
What is sine, actually?
Forget triangles for a second. Imagine you're standing at the center of a circle with radius 1 (called the "unit circle"). You walk along the edge of the circle, starting from the rightmost point. At any moment, your height above the center — that's sine. Your horizontal distance from the center — that's cosine.
That's it. Sine is just "how high up are you on a circle."
The triangle stuff is a consequence of this, not the definition. If you're at some angle θ on the circle, draw a line straight down to the horizontal axis — you've made a right triangle. The vertical side length equals sine(θ). That's why it shows up in triangles. But the circle is more fundamental.
Your first question: why connect the dots?
This is a legitimately deep question and most math teachers would fumble it. You're right that knowing sin(0) and sin(30) doesn't automatically tell you what sin(12) is. You could imagine all kinds of wild behavior in between.
The reason we can connect them smoothly is that sine comes from the circle, and moving around a circle is continuous — there's no jumping, no teleporting. As the angle changes by a tiny amount, your height changes by a tiny amount. You're not measuring some arbitrary collection of facts; you're tracking a point moving smoothly around a curve.
So it's not an assumption — it's a consequence of what sine actually is.
Your second question: why curved, not straight lines between 0° and 90°?
Because height-on-a-circle doesn't change at a constant rate as you walk around.
Think about it physically: when you're near the bottom of the circle (angle near 0°), moving a little bit along the edge changes your height a lot — you're climbing steeply. When you're near the top (angle near 90°), moving along the edge barely changes your height at all — you're moving almost horizontally.
So sine starts out rising quickly, then slows down as it approaches 1. That's the curve. A straight line would mean "rising at a constant rate," which would only happen if the circle were somehow a straight line — which it isn't.
The questions you asked at 14 or whenever this was are essentially the questions that lead to calculus. "How fast is it changing right now?" is exactly what a derivative answers. You were thinking like a mathematician and your teacher probably didn't know what to do with that.