How the "anonymous Gothamite" would "trigger the bomb"?

To answer your questions in order...

The triggerman allegedly has the means with which to detonate the bomb. This may be one of any number of methods, of which a remote control is just one.

Bane said (in your quote) that anyone trying to leave Gotham, or any interference from outside Gotham would result in the bomb being triggered. This is why the army are blocking the bridge that Gotham citizens could otherwise escape across - if nobody can leave, then so long as the army stay outside Gotham, then the triggerman doesn't detonate the bomb.

Gordon's an experienced police officer and a cynic. He's built to question everything and everyone, and analyse the information he's given. His judgement is that Bane is bluffing, but not everyone will think the same way and this will cause a large number of the population to believe it. This can breed mistrust among the citizens as people will not be able to tell who the mystery person working for Bane is. This helps to throw off any attempts to organise people together to resist Bane.

The point of giving someone random the detonator is to obscure who has it, and therefore make it all but impossible for anyone with designs on resisting Bane's authority to take out the person with the detonator. Bane can presumably order the person to set it off, or would have told said person to set it off under certain circumstances. It induces fear in the public to prevent them trying to rise up.

I think the answer to your 2nd question is actually quite simple:

As we find out at the end of the movie Bane was really working for Thalia al Ghoul (Miranda), the whole plot to destroy Gotham was "her" plan - or rather the plan of her father which she conntinued to carry out. And after Batmans last fight with Bane in the movie, we see Thalia stab Batman, tell him who she really is and than she uses detonator for the bomb, luckily by this time Gordon diabled the remote trigger on the bomb. So Bane wasn't lying about some one else having the trigger - the only thing is at that point no on knew that Bane wasn't the mastermind of the plan.

As for your 1st question:

During the movie Bane speaks the truth most of the time, so why lie when the truth is so devestating already. I also don't think that Bane concerned himself with the fact that some people might not belive him about the triggerman.

It was a ploy to make people afraid to do anything. If you have no idea who holds the trigger, and the bomb will be triggered if you don't follow the rules, you won't try anything to interfere because there's a possibility that you will be killed.