Killing a Animal to prevent painful of that Animal.Is it consider as a bad thing for me according to buddhism?

is it a bad thing for me

Do you believe the karma system?

If you do not believe it, then it is neither good/right nor bad/wrong according to Buddhism. This is not a topic that Buddhism can address, just like Buddhism cannot address how to heal burning. In fact, this may be the answer. You do not need to go to Buddhism for an answer. Just go back to common sense. In real life, a lot of problems can be solved by common sense.

If you believe the karma system, it is not a good thing to the dog. The dog is suffering because of the fire, but what drives it to the fire and the suffering is its karma. The dog will have a new life if you killed it, and continues suffering. Unfortunately you can rescue it from the suffering of burning, but not rescue it from the suffering. Even Buddha can not rescue the dog from suffering. However, in many lives later, the dog may bore as a human, and may learn about Buddhism. And since it is not a good thing to the dog, it is not a good thing to you.

I am not sure if it is a bad thing. I guess it may not be a bad thing as long as you kill the dog solely for free it from suffering, not as an excuse to fulfill your desire for killing, or an excuse to stop yourself feel uncomfortable or an excuse to stop your helplessness (the eager that "I have to do something"). You may need to examine your mind carefully via, say, mediation, since desires sometimes hide deeply.

Also, it is dangerous if you are not a butcher already. You need to find a proper way to kill the dog. Some people get used to killing or even get excited after they actually kill an animal.

Also, do not think too much about whether it is good or bad for you. The bad thing is dog has been burned. And it is bad for the dog. And it already happens. If you can heal it, then go ahead. If you cannot, then you may ask someone else's help. If neither of the above applies, then it is a difficult choice, because the situation is difficult. And Buddhism cannot let you choose among difficult choices blindly. It is difficult for the dog, and it is difficult for you. The Buddhism cannot help much here, since the very assumption of Buddhism is life is suffering, and life is difficult.

Buddhism is not karma economics trying to maximizing good karma and minimizing bad karma. Otherwise you may get illogical conclusion like killing the dog violently may speed up its process to become human again. Buddha gave some examples of karma, which are just illustrations to help you understand the system. Buddha is teaching relief, not karma calculation. Good karma is good because it drives you towards relief. You should not try to pursuit good karma. It is not the goal of Buddhism. In fact the ultimate goal of Buddhism is not be ignorant about karma to avoid producing new karma, and eventually get rid of the karma system after paying off all existing karma.

Some verses from the Dhammapada;

129. All tremble at violence; all fear death. Putting oneself in the place of another, one should not kill nor cause another to kill.

130. All tremble at violence; life is dear to all. Putting oneself in the place of another, one should not kill nor cause another to kill.

Though a dog might fear pain, is it possible they'd fear death (or a person's violence resulting in their death) even more?

Buddhism doesn't recommend "mercy killing". It's considered very wrong (unforgivable) for a monk to even recommend the alleged "advantages" of suicide, to someone who's in pain.

Even mercy killing is considered bad and also breaks the precepts.

The dog is going through its pain due to karma. Say you kill it and goes to a worst hellish state by killing it you are putting to a far worse experience.

If you treat the dog's wounds and try to heal it you are creating good karma for yourself and also easing the pain for the dog. This is be best avenue to consider.