The bent nail is probably a result of you losing the nail earlier through trauma damage..
The nail IS thickening by the look of the pics so more than likely it could be a fungal infection - get it treated as it can spread to other toenails.
Your nails cannot bleed as they have no blood vessels. It's more likely that you are cutting the nail too short, and are catching the skin just below the nail plate. When using the type of cutters you describe they can pull more nail into the scissor blade than you expect.
So, be careful, try not cutting as short, and if you cause them to bleed again use some antiseptic on the bleeding area.
Your nail looks to be a bit ragged and uneven, so try gently filing them after you've cut them, so that they're smooth and less likely to snag on socks/tights etc.
Finally, if you find it difficult to cut your nails yourself, see if you can go see a podiatrist on a regular basis and let them do it. They can usually spot things that you can't, and lets face it, if your feet hurt, then you hurt everywhere.