Are the names Akiva and Ya'akov related?
Another name probably related to these is עוקבא Ukva (borne by several people in the Amoraic and Geonic eras).
This site lists only one proto-Semitic root עקב (although, as usual, the various descendant languages extend it in different senses). So it's most probable that Akiva is indeed based on the same root as Yaakov, and means something like "heel-grabber" or "follower" or something similar.
It's also worth noting that Targum Yerushalmi to Deut. 25:9 (cited in Jastrow's dictionary) uses the word עקיבא with the meaning of "heel."
They are related mystically. Arizal states that Akiva was a gilgul of Yakov Avinu. One of his supports are the similarity in name. Additional point he makes is : both married a woman named Rachel.
He says that Yakov needed to atone for the 22 years he was away from his father Yitzchak. Akiva accomplished this by serving his Rebbe, Nachum Ish Gamzu, for 22 years.
I'd strongly assume so (both spelled with the root עקב ). There's also the Mishnaic name "Akavia" עקביה, most likely also related.
Note also that Rabbi Akiva's first love is named Rachel -- as was the Biblical Ya'akov's.
The Mishna is full of nicknames and derived names (e.g. Yosef -> Yosei -> Issi); R' Avigdor Miller is quoted as saying that even the less-Hebraic-sounding names of the Gemara (Ashi?) were nicknames or derivatives of Hebrew names.