Today, I killed

Try to grab a snake by its mouth. It will bite you. Your wrong grasping of Dhamma can land you in many moral dilemmas and can create existential crisis. The precepts are meant for those who have seriously committed to the 8 fold path. Being a householder you have to take care of yourself , your family , your society etc. You can’t just sit worrying about killing an insect like this. Karma involves lot of common sense.Does killing an insect make you a bad person ? Is it your hobby to kill insects ? Or do you kill insects for the welfare of the society ? Clearly your intentions are good and you are not a bad person. Killing one insect won’t land you in any trouble. If you lie or steal or kill animals as a monk then you have serious problem at hand because it goes against the teachings of Dhamma and you will fail in your endeavour to liberate from samsara...

There is no comparison you kill the poor insect, than you kill your own soul. The soul is guide, own light, immortal. When you forget soul, you are in pain. Its not a wonder that one's body dies, but a great wonder that they think soul dies. Their soul is already dead. Shruti proclaim, 'Soul is immortal, my dear', its your guide, inner master, guide and refuge. Whoever awakes to the inner refuge, act of killing and sense of being killed by another is mere a phantom. But not in your case, as your soul already dead, so karma tends to be there. Don't fall trap to Buddhist lore. They cling to scriptures. Give up that which is impermanent, which is not yours, you will see your own light. All suffering fades away.

Buddha said: Self is refuge, self is inner light and guide. Upanishada says: Thou are that! Many awakened ones still existed in Indian land, they don't bound to particular religion. They live in isolated places in Himalayas. They beg food and give instructions only to those who are interested in it. Whoever find his inner refuge, all relationship ends right there, all properties flung aside, he trespasses from trap of good and evil, he lives in solitude with one with light. How can his happiness be compared?

What do you think of this story? Was I wrong the first time? The second?

First time: no kamma ( unwitting/unintentional action ).

Second time: mixed kamma with bright and dark results ( active intention/action to kill + active intention to alleviate the state of prolonged suffering )

If so, how can I mitigate this bad kamma?

Be more mindful of whatever action you'll do at your desk and away from your desk in the future.

And finally, how can we help him have a better rebirth, if at all possible?

While there's no guarantee of benefit, definitely can't hurt to radiate your Metta/Karuna with sutta reciting and more mindful conducts thru the three gates of body/speech/mind.

Wrong second time. First time there is no factor of intention, second time you wanted him to die for whatever reason.

Take a hypothetical example of a person who is unable to kill another intentionally, it being not in his range he couldn't have done it. If not killing is classed categorically as good then the person who doesn't kill can not be blamed for not kiling in any circumstance.

It can thus be inferred by simple logic that the correct course of action with these premises is not killing in any circumstance. It does not require further analysis because it's such a simple equation in this context.

Therefore if one holds that wishing others to die is bad then the guideline is clear.

If one holds that compassionate killing is the lesser evil or is a good then one holds that killing is sometimes justified and that's a whole different story requiring a demonstrable basis for that assertion, as it goes against the norm of killing being bad it is then an extraordinary claim and assumes the burden of proof. If no proof then it's just a view that goes against the norm and what is otherwise held to be true.

The second?

I met a pigeon this year which was crippled with an illness.

I gave it shelter, food, and water, for about 10 days until it died.

I figure that was maybe kinder than trying to kill it when it was helpless.

If so, how can I mitigate this bad kamma?

"Stop doing it" -- see this answer which quotes SN 42.8.

And finally, how can we help him have a better rebirth, if at all possible?

Is "make a better/kinder world for him to be reborn into" a valid answer?

What do you think of this story? Was I wrong the first time? The second?

I think these questions are displaying concern for what others might think. There's a risk that it contributes to throwing us off balance:

Now, gain arises for a well-instructed disciple of the noble ones. He reflects, 'Gain has 
arisen for me. It is inconstant, stressful, & subject to change.' He discerns it as it 
actually is...

Loss arises... Status arises... Disgrace arises... Censure arises... Praise arises... 
Pleasure arises...

If so, how can I mitigate this bad kamma?

You can't mitigate bad karma. What you can do is try to be equanimous, and prevent the risk for more bad karma in the future.

...Pain arises. He reflects, 'Pain has arisen for me. It is inconstant, stressful, & subject 
to change.' He discerns it as it actually is.

 His mind does not remain consumed with the gain. His mind does not remain consumed with 
 the loss... with the status... the disgrace... the censure... the praise... the pleasure. 
 His mind does not remain consumed with the pain.

https://www.accesstoinsight.org/tipitaka/an/an08/an08.006.than.html

And finally, how can we help him have a better rebirth, if at all possible?

If we agree that rebirth pertains to the notion of a self, it won't be possible because it's already dead, and besides, it's debatable whether animals actually have a perception of a self to begin with.

However, whether animals have a self or not is irrelevant for whether we should act based on the brahmaviharas or not.

(See this answer, for instance: https://buddhism.stackexchange.com/a/38/11699)