Mtg question does killing a creature counter it's ability?

Short answer: Lorthos' ability will resolves.

Detail to understand why it does so you can tell your friends they were wrong:

You attack with Lorthos and this triggered Lorthos' ability. You have a choice to pay 8 or 0 mana. Lorthos' ability is put on the stack whether you pay or not. Assume you paid 8 mana. Your opponent responses with Doom Blades targeting Lorthos. Doom Blade is put on the stack on top of all other objects already in the stack. So we have Doom Blades as the top most item on the stack and Lorthos' ability on the bottom. Assuming no other player cast spells or activate abilities. The stack resolves from the top, in this case, Doom Blade. When Doom Blade resolves, Lorthos' is destroyed. Next, Lorthos' ability resolves. As it resolves, the game checked that you paid 8 mana, so you tap up to 8 target permanent. If you didn't pay, then there is no more action to perform. Lorthos' ability still resolves; it just does nothing.

Comprehensive Rules 112. Abilities

112.7 The source of an ability is the object that generated it. ... The source of a triggered ability (other than a delayed triggered ability) on the stack, or one that has triggered and is waiting to be put on the stack, is the object whose ability triggered.

112.7a Once activated or triggered, an ability exists on the stack independently of its source. Destruction or removal of the source after that time won't affect the ability ...

Note: Lorthos' ability use "MAY pay 8 mana". If you forgot to pay, then you cannot pay later. If you do not pay (whether you forgot to pay or not having enough mana to pay or choose not to pay) when the ability is triggered, you lost the only window of opportunity to do so. There are cards with triggered abilities that you HAVE to do. Example: Soul Warden triggered ability says "Whenever another creature enters the battlefield, you gain 1 life." You cannot not gain 1 life with this ability. Not gaining a life is NOT an option and you don't have any choices here.

Once an ability goes on the stack it is independent of the source so removing the source does nothing. The only ways to stop an ability is to counter it with cards like Stifle or Voidslime, or to remove the target.

Killing/removing a permanent will not negate/counter its activated or triggered abilities.

The analogy to use is imagine someone throwing a spear at you. Just because you killed the guy who threw the spear, it didn't stop the spear from continuing its path into hitting you.