The proper sentence for the second one would be: "Bugs annoy me all the time."

I wouldn’t say ‘irk’, unless I wanted to sound slightly pretentious : ‘It irks me that I have to waste time dealing with all these bugs’. It’s more usual to say ‘annoys’, and of course there are other, more vulgar, ways of saying this.

As Spencer hinted at, BUG can also be a verb meaning to IRK. It's used more commonly. He's so annoying. He really bugs me.

Yes, “irk” is a normal word that can be used in daily conversation.

Bugs bug you! I’ve never heard irk used for bugs. It sounds a bit strange, but someone would understand you if you said that. Irk the verb and irksome the adjective are rare but would be widely understood. I think people and the things they do are more commonly irksome. This may be a case where there is no formal rule but use is dictated by norms that can’t really be defined.