What's this mecha anime with a robot world championship?
OK I think I am pretty sure the anime you are looking for is Daigunder



POINT 1: The main character is Akira Akebono and his grandfather, Hajime Akebono, is a famous research scientist who created the robots. He is like Ash Ketchum except with robots instead of pokemon. The robot which says ,"I have the sticker inside me too?"  is Daigunder units' human like AI robot called Ryugu, flaming robot warrior. 

POINT 2: Coming to the cobweb thing, there is an episode where the robot malfunctions and tries to harm Ryugu even after the battle is over. Akira jumps in (like Ash saving Pikachu or hugging it to death ) and tries to protect Ryugu. I think they get disqualified for a that tourney. But Akira gains Ryugu's trust. 

POINT 3: And the robot cutting wood must be Bulion, the blue thrasher. Its just a small scene in training. I don't clearly remember which episode. 

POINT 4: The girl in question is Haruka, she at first thinks Akira is a dumb kid (as he really is). But eventually joins this team as a manager.She gets a job at her dream company I guess but instead, joins this team.

check this out for a list of episodes. But all you points come up within 4 to 5 episodes of the series. Hope this helps :D
https://www.google.com/search?q=daigunder+episode&amp;ie=utf-8&amp;oe=utf-8#q=daigunder+episodes