An optional effect will not even trigger if the event that it triggers off of happens in the middle of a chain. This is clearly spelled out in numerous "Missing the Timing" rulings.
Heart of the Underdog
Missing the Timing: Suppose you have 3 copies of "Heart of the Underdog" on the field, and draw a Normal Monster Card during your Draw Phase. All 3 copies of "Heart of the Underdog" immediately activate their effects, and because they activate simultaneously, they form a chain with Chain Links 1, 2, and 3. If you draw a Normal Monster card for Chain Link 3 or Chain Link 2, your copies of "Heart of the Underdog" do not activate again. This is because they are "when... you can" optional Trigger Effects, and you "miss the timing" because you drew the Normal Monster Card during a chain and not as Chain Link 1. However, if you draw a Normal Monster Card for Chain Link 1, all 3 copies of "Heart of the Underdog" will activate their effects again.
Once an effect has been successfully triggered and placed on the chain, there is no missing the timing. You can only miss the timing for the activation of an effect.
Peten the Dark Clown
Missing the Timing: "Peten the Dark Clown" is a "when... you can" optional Trigger Effect. This means that "Peten" being sent to the Graveyard has to be the very last thing that happened in order to activate its effect.
If "Peten the Dark Clown" is Tributed for a Tribute Summon, the last thing that happened was that a monster was Summoned, so the effect of "Peten" does not activate.
If "Peten the Dark Clown" is Tributed for "Mystik Wok" or "Cannon Soldier", the last thing that happened was that those effects resolved, so the effect of "Peten" does not activate.
If "Peten the Dark Clown" is destroyed in a chain and not as Chain Link 1, his effect does not activate. (Example: "Pot of Greed" is activated and "Ring of Destruction" is chained, targeting "Peten".)
If "Peten the Dark Clown" is destroyed by "Cyber Jar", the last thing that happened was that monsters were Summoned to the field, so the effect of "Peten" does not activate.
If "Peten the Dark Clown" is discarded as a cost for "Tribe-Infecting Virus" or "Raigeki Break", the last thing that happened was that those effects resolved, so the effect of "Peten" does not activate.
If "Peten the Dark Clown" is discarded because of "Morphing Jar" or "Card Destruction", the last thing that happened was that both players drew cards. So the effect of "Peten" does not activate.
Assault on GHQ
If you destroy "Peten the Dark Clown", you miss the timing on its activation because it was not the last effect to resolve, so you cannot Special Summon another "Peten the Dark Clown".
"Missing the Timing" is all about missing your chance to activate an effect, not resolve it.