Force, Work, and Energy

To reinforce this concept, if I'm given that a particle is sliding down the path y=x^2 from a height of 4 to a height of 0 in a uniform gravitational field F=<0,-mg>, then it doesn't matter if I parametralize this as r(t)=<t,t^2>, even though the particle wouldn't really fall at that rate?

Yep, thats the idea, just remember to state what values of t you're using to integrate. For r(t)=<t,t^2> from (2,4) (0,0) you should say t goes from 2 to 0. Not entirely sure a gravitational field is a good example, since g normally is a function of time, but if you're treating it as a constant, go for it.

Let's say that a particle is sliding down an irregular ramp. Since friction is F= -μFn, then would the energy lost due to friction be this?

E= -∫ μF • N ds

The energy loss due to friction is equal to the work done against it, yes.

E= -∫ μF(t) • T'(t)/||T'(t)|| ||r'(t)|| dt

I'm not sure about this, because I've never taken vector calculus before, but I do know that doing this process over a over a surface gives the flux，or the amount of liquid that passes through the surface. My equation seems to be correct for the frictional force, but because of flux, the equation also seems to be counter-intuitive. Also, is there any way to simplify N ds? It seems really messy.

N ds can be simplified, but it depends entirely on the physical system you're looking at. For example, a circle around the origin in the x-y plane, would have a normal vector either in the +z direction, or -z direction, so you could just stick a unit vector (+k or -k) instead of N, and then the equation becomes pretty easy.
There isn't a mathematical method to simplify it in general that I can think of though

Could you please elaborate on this? How would you use

W= ∫ F • dl = ∫ F cos(θ dl

if θ isn't constant? I thought this equation only works when you're dragging a box on a flat surface and the force or angle the force applied isn't changing.

If theta isn't constant, then its a horrible horrible equation. You either need to express dl as a function of theta, or express theta as a function of length. Both are doable, but its very dependent on any given situation as to how you should go about it.