"focal point", "image point" - careful with the terms.

All you need to worry about is where the light is focused in the smallest space. Yes, that should be pretty bright. Yes, you could use a business card to detect it - just figure out where the image is smallest.

Now, how to capture more light. Look at this image:

http://upload.wikimedia.org/wikipedia/commons/7/71/Lens3.svg

As the object moves further away from the lens and to the left, the image also moves to the left, closer to the lens. When S1 = infinity, then S2 = f, and that's the closest the image will get to the lens, and the image is smallest at that point. However, because the source is so far away, the image may be small, but it's pretty faint.

So yes, the answer is: use a more convergent lens (a stronger lens) because then the source could be closer to it. Also, a bigger lens might help. There's a trade-off how far to place the source; if it's too close, you capture a lot of light, but the image is large; if it's too far, the image is small, but you don't capture much light. At S1 = 2f, the image is about the same size as the source.

It might help to enclose the source in some sort of box, the inside should be either bright white (good), or reflective (better), or reflective with a shape that focuses light back into the source (best). The size of the lens becomes somewhat less important in this setup.

This is a hard problem, don't be surprised if there are no clearly defined answers to it. Unless the source is equal to or smaller than the fiber entrance, then it's somewhat easier - put a reflective sphere around the source, poke a hole, put a ball lens on the hole and couple the sphere with the entrance of the fiber.

http://www.edmundoptics.com/technical-support/optics/understanding-ball-lenses/

A tiny LED chip is easy to couple with a fiber aperture. A large fuzzy lamp is much harder.