I can't claim to avoid attachment, maybe I try to avoid clinging since all things are impermanent. I am still in a dualistic mindset, clearly! If another sentient being such as a pet is in distress which cannot be removed or ameliorated, allowing a vet to end that suffering can be the greatest act of compassion possible. I have been in that position, the right choice is based on love, not on a selfish desire. Buddhism is said to fly with two wings - compassion and wisdom.
Here's my problem with impermanence. The theory completely falls apart when you say "Impermanence is Permanent" (Zaim Alzzalam)
It's perfectly healthy to suffer through grief from the passing of a pet. It's not a one day thing, it takes days, weeks, even months before your mind/body begins the healing process.
Not at all. Impermanence is permanent, obviously, so therefore there is no such thing as impermanence. That something is unstable (what isn't?) has no bearing on permanence or impermeanence.