1.If you want something that's close to your mockup I'd use particles (It doesn't have to be a fully blown particle system).
Render your particles in the form of the polygon on a RenderTexture. Make sure to use additive blending on the particles. The particles inside of the polygon will smoothly blend into each other while the particles on the outside will give the soft edge that you want. (An example of the effect can be watched in this youtube video: Additive Particles Video
Now render the RenderTexture onto your main screen and you're done. The RenderTexture is necessary so that the particles don't blend with your background.
You can try to put the triangles directly onto the particle texture and see how this works out. Otherwise render them on top of your "particle soup" as a separate layer.
Created a quick mockup in an updated jsfiddle that looks like this
You can find the updated demo here
2.Each particle has a velocity and an origin. When your player touches the polygon, you change each particles velocity proportional to the players velocity. The farther a particle is away from your player, the less it is affected by the players velocity.
The formula to calculate a particles velocity would be something like this:
//player.velocity and particle.velocity are vectors
//k is a factor to enhance or weaken the influence of players velocity
var distanceToPlayer = (player.position - particle.position).length();
particle.velocity = particle.velocity + ((k * player.velocity) + particle.velocity) * (1/distanceToPlayer);
To calculate the position of the particle you put this in your update method:
var speedY = -(springConstant * (particle.position.y - particle.origin.y)) - (dampingFactor * particle.velocity.y);
var speedX = -(springConstant * (particle.position.x - particle.origin.x)) - (dampingFactor * particle.velocity.x);
particle.position.y = particle.position.y + speedY;
particle.position.x = particle.position.x + speedX;
particle.velocity.x = particle.velocity.x + speedX;
particle.velocity.y = particle.velocity.y + speedY;
This should give you a "fluid" where each particle swings around its origin when the player stirs the fluid. The springConstant changes how much a particle swings away from its origin and the dampingFactor how fast the particle comes to rest.
You may have to tweak the code since its the modified version of a 1d simulation I use in my game.
Now with a demo: Demo
Just tweak the 3 constants at the top until the fluid behaves like you want it to.