user12857: At work, a tiny little flying insect landed on my desk. At first I thought he was dead because he wasn’t moving, so I gently pushed him to see if he reacted, and he moved so I left him alone. Some time later, when I picked up a call, I shoved my electronic cigarette and it unfortunately crushed him. So I thought I killed him unwittingly, I felt guilty. I took a closer look, and I could see that although crushed his antenna was still moving, probably in agony. I felt bad, now right away a being was suffering and was going to die because of me, was perhaps afraid? I decided to crush him for good, voluntarily this time, to alleviate his suffering.
What do you think of this story? Was I wrong the first time? The second?
If so, how can I mitigate this bad kamma?
And finally, how can we help him have a better rebirth, if at all possible?

user2282: 
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.

user-1: 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.

user254: 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?

user-1: 
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)

12857: <p>At work, a tiny little flying insect landed on my desk. At first I thought he was dead because he wasn’t moving, so I gently pushed him to see if he reacted, and he moved so I left him alone. Some time later, when I picked up a call, I shoved my electronic cigarette and it unfortunately crushed him. So I thought I killed him unwittingly, I felt guilty. I took a closer look, and I could see that although crushed his antenna was still moving, probably in agony. I felt bad, now right away a being was suffering and was going to die because of me, was perhaps afraid? I decided to crush him for good, voluntarily this time, to alleviate his suffering.</p>
<p>What do you think of this story? Was I wrong the first time? The second?</p>
<p>If so, how can I mitigate this bad kamma?</p>
<p>And finally, how can we help him have a better rebirth, if at all possible?</p>
 2282: <blockquote>
<p>What do you think of this story? Was I wrong the first time? The second?</p>
</blockquote>
<p>First time: no kamma ( unwitting/unintentional action ).<br/><br/>
Second time: mixed kamma with <a href="https://suttacentral.net/mn57/en/bodhi" rel="nofollow noreferrer">bright and dark results</a> ( active intention/action to kill + active intention to alleviate the state of prolonged suffering )</p>
<blockquote>
<p>If so, how can I mitigate this bad kamma?</p>
</blockquote>
<p>Be more mindful of whatever action you'll do at your desk and away from your desk in the future.</p>
<blockquote>
<p>And finally, how can we help him have a better rebirth, if at all possible?</p>
</blockquote>
<p>While there's no guarantee of benefit, definitely can't hurt to radiate your <a href="https://en.wikipedia.org/wiki/Brahmavihara" rel="nofollow noreferrer">Metta/Karuna</a> with <a href="https://www.accesstoinsight.org/tipitaka/kn/snp/snp.1.08.amar.html" rel="nofollow noreferrer">sutta reciting</a> and more mindful conducts thru the three gates of body/speech/mind.</p>
 -1: <p>Wrong second time. First time there is no factor of intention, second time you wanted him to die for whatever reason.</p>
<p>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.</p>
<p>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.</p>
<p>Therefore if one holds that wishing others to die is bad then the guideline is clear.</p>
<p>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.</p>
 254: <p><strong>The second?</strong></p>
<p>I met a pigeon this year which was crippled with an illness.</p>
<p>I gave it shelter, food, and water, for about 10 days until it died.</p>
<p>I figure that was maybe kinder than trying to kill it when it was helpless.</p>
<p><strong>If so, how can I mitigate this bad kamma?</strong></p>
<p>&quot;Stop doing it&quot; -- see <a href="https://buddhism.stackexchange.com/a/39468/254">this answer</a> which quotes SN 42.8.</p>
<p><strong>And finally, how can we help him have a better rebirth, if at all possible?</strong></p>
<p>Is &quot;make a better/kinder world for him to be reborn into&quot; a valid answer?</p>
 -1: <blockquote>
<p>What do you think of this story? Was I wrong the first time? The second?</p>
</blockquote>
<p>I think these questions are displaying concern for what others might think. There's a risk that it contributes to throwing us off balance:</p>
<pre><code>Now, gain arises for a well-instructed disciple of the noble ones. He reflects, 'Gain has 
arisen for me. It is inconstant, stressful, &amp; subject to change.' He discerns it as it 
actually is...

Loss arises... Status arises... Disgrace arises... Censure arises... Praise arises... 
Pleasure arises...
</code></pre>
<blockquote>
<p>If so, how can I mitigate this bad kamma?</p>
</blockquote>
<p>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.</p>
<pre><code>...Pain arises. He reflects, 'Pain has arisen for me. It is inconstant, stressful, &amp; 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.
</code></pre>
<p><a href="https://www.accesstoinsight.org/tipitaka/an/an08/an08.006.than.html" rel="nofollow noreferrer">https://www.accesstoinsight.org/tipitaka/an/an08/an08.006.than.html</a></p>
<blockquote>
<p>And finally, how can we help him have a better rebirth, if at all possible?</p>
</blockquote>
<p>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.</p>
<p>However, whether animals have a self or not is irrelevant for whether we should act based on the brahmaviharas or not.</p>
<p>(See this answer, for instance: <a href="https://buddhism.stackexchange.com/a/38/11699">https://buddhism.stackexchange.com/a/38/11699</a>)</p>
