I could have eaten this and...
 

I could have eaten this and thrown the rest away
I could have eaten this and given the rest to you
I could have eaten this and gone to bed

After Podría haber comido esto........., do I use another verb “haber” to finish the second clause, or do I use an infinitive verb to finish the second clause?

Podría haber comido esto
1. y haber tirado lo demás, y haberte dado lo demás, y haberme acostado

or

2. y tirar lo demás, y darte lo demás, e irme a acostar

or some other construct that I'm not considering.

Any and all explanations and/or examples are greatly appreciated.

Let's see what Angelica or Rusty has to say to confirm this, but I would say,podria haber comido esto y botar (tirar) lo que queda. If the sentence began with if the past subjunctive must be incorporated in the verb following if.
Si pudiera haber comido esto, botaría lo que queda.

@Bobbert: I think someone else will have a more technical explanation, but for me, both proposals work.

1. If you repeat "haber" is fine because the conjugation agrees with the original tense of the action.
- Podría haber(me) comido esto y haberte dado lo demás / y haber tirado lo demás / y haberme acostado.

2. If you don't repeat "haber", the sentence will sound better to most ears, because Spanish speakers aren't fond of repetition of something that is already understood. ;)
- Podría haber(me) comido esto y darte lo demás / tirar lo demás / acostarme (irme a acostar).

@Poli: "Botar" is not a common verb in Mexico, but it does work for other regions. :)
In your example with the conditional, you would need a second verb with a compound tense and the third verb in preterite, because "haber + -ado/-ido" are actions in the past:
- Si pudiera haber comido esto, habría botado lo que quedaba. (You weren't capable of eating a certain amount of food, but if you could have, you would have thrown away what was left of it.)

But if your conditional is used for a situation in the present, you don't need compound tenses:
- Si pudiera comer esto, botaría lo que queda. (You don't feel capable of eating a certain amount of food, but if you could, you would throw away the rest.)

Thank you for taking the time to respond. That clears it up for me.