Can I use the pass for the Asahikawa / Asahidake Onsen bus as well? If not, Lonely Planet tells me that the Asahidake Onsen / Asahikawa ticket is free if you spend more than X yen in Asahidake Onsen?

Can you confirm the bus times, i.e. is http://www.japan-guide.com/e/e6776.html up to date?

Do the JR Hokkaido or JR East passes or both cover the journey Hakodate to Aomori?

The regional JR passes sound nice, but very often they end up to not be the best bet.

Consider car hire if you want to really explore Hokkaido. Soooo worth it.

From your response it looks like I would need both the JR Hokkaido and JR East passes to cover the journey from Hakodate to Aomori. I am still confused though because if you read the conditions of both the JR Hokkaido and JR East passes, neither seems to cover the journey from Hakodate to Aomori, and buying both would not solve the issue?

Is the JR Hokkaido pass valid on Hokkaido Railway Co. Lines, i.e. is Hokkaido Railway Co. Lines the same as JR?

Any view on the other questions?

That is because the Hakodate to Aomori trip crosses company borders, so each pass is only valid up to its company's respective border station. Have you considered just purchasing a regular ticket for the Hakodate/Aomori portion? It costs 4830 yen per person by JR Hakucho Limited Express train.

Is the JR Hokkaido pass valid on Hokkaido Railway Co. Lines, i.e. is Hokkaido Railway Co. Lines the same as JR?

Yes

http://en.wikipedia.org/wiki/Hokkaido_Railway_Company

Can you confirm the bus times, i.e. is http://www.japan-guide.com/e/e6776.html up to date?

With a Japan Rail Pass [JR Pass], you can go from Hakodate to Aomori on one Ltd. Express train. If neither a Hokkaido Rail Pass nor a JR East Pass covers the fare or the charge, you can hold down an additional payment to 1,970 yen per adult by changing trains and choosing a non-reservation space (free of charge) on a Ltd. express train, but it costs about 1 or 1 + 1/2 hour longer.
Comparison:
- Hakodate 11:28 Ltd. Express Hakucho #20 - 13:36 Aomori vs.
Hakodate 10:12 Local, Kikonai 12:08 Ltd. Express Hakucho #20, Kanita 13:47 Local, 14:26 Aomori.
- Hakodate 15:42 Ltd. Express Super Hakucho #30 - 17:37 Aomori vs.
Hakodate 14:44 Local, Kikonai 16:19 Ltd. Express Super Hakucho #30, Kanita 17:30 Local, 18:11 Aomori.

:buying both would not solve the issue?

I presume that's right.
The border between the areas of the two companies is in Aomori Prefecture.
Kikonai - Nakaoguni: Tsugaru-Kaikyo Line of JR-Hokkaido;
Nakaoguni - Kanita - Aomori: Part of Tsugaru Line of JR-East.
However, passengers can not change trains at the border, because the trains running via the border pass the border; what is more, all the trains are Express / Ltd. Express trains.
When a passenger uses a Japan Rail Pass [JR Pass] / a Hokkaido Rail Pass / a JR East Pass, a railroad cost which is not covered by the Pass should be paid separately.
So, if a passenger crosses the border on such an Express / a Ltd. Express train, presumably neither a Hokkaido Rail Pass nor a JR East Pass covers the cost even when used together, because the cost can not be paid separately for one company's part.

//

No, it is not a JR bus.

If not, Lonely Planet tells me that the Asahidake Onsen / Asahikawa ticket is free if you spend more than X yen in Asahidake Onsen?

That used to be like that. But these days, I think that everybody has to pay always if taking the bus.

Can you confirm the bus times, i.e. is http://www.japan-guide.com/e/e6776.html up to date?