No.

According to this rule change, the horse may compete if the bleeding has resolved later in the day. 

This depends on the circumstances, which must be evaluated by the judge or the ground jury. If the source of the blood is determined to be caused by rider/handler or equipment, elimination is mandatory. Whether or not the area is actively bleeding is also a factor in how the judge may determine the most appropriate outcome.

They will rely on their experience to assess the whole picture. If the blood is coming from the mouth, the judge or ground jury will want to know if the source is the horse biting their tongue or cheek, or if the blood is induced by either the bit or the use of the bit. They will examine the bit construction and fit and will consider what the athlete was doing with their hands just before the blood appeared. They will also look at the horse’s mouth and tongue to see if the source is visible. The athlete will have the chance to wipe the horse’s mouth and see if the bleeding continues. The expectation is a minor cut from biting the tongue or cheek will stop bleeding quickly. 

In the case of blood elsewhere on the body, the official will also assess the location relative to the athlete/equipment, the use of the equipment and aids by the athlete and use the complete picture of the situation to determine the source of the blood. 

No.

They will be asked to continue from the point at which they were stopped.

In classes with more than one distinct phase, such as a round and a jump-off or a work-off, a horse who is eliminated after the first phase will be placed according to the score they accumulated in the first phase, prior to elimination.  
 
For example, if four jumpers complete an initial course double clear and move on to the jump-off, and one horse is eliminated for blood in the jump-off, that horse will be placed based on their performance in the first round, meaning they would be placed fourth overall (last of the group that moved to the jump-off).

General Rules are the base, but in any situation where a breed or discipline rulebook has stricter or additional requirements in a subject area, those breed/discipline rules will also apply. In the case of the blood rule, for example, the rule for Dressage requires that the presence of blood anywhere on the horse for any reason results in elimination; that will continue to apply for Dressage participants.