1) I seriously do not recommend taking apart and swapping parts between lithium-ion batteries. They're very temperamental about how they're charged, and mis-charging them can cause them to catch fire or explode. If you do wish to try, consult with an expert who fixes laptop batteries for a living.
2) Lithium-ion cells are 3.6-3.85 Volts. The 11.1 V battery is just parallel groups of 3 cells arranged in series (3.7*3 = 11.1). Same with the 10.8V (3.6*3 = 10.8). There's no way of knowing if the difference (3.6 vs 3.7 V) is an actual difference in the chemistry, or just a difference in the specs on paper. Yet another reason not to try fixing this yourself.
3a) The Wh refers to the capacity of the battery. The laptop doesn't care about the capacity. In all likelihood the laptop will work fine at either 10.8 or 11.1 Volts. So if you can find a battery rated for that voltage with any Wh capacity which fits, it will most likely work. (The charging mechanism determines the charge state of the battery by measuring the voltage, which slowly increases as the battery is charged. So Wh capacity doesn't matter.)
3b) Based on the processor, this laptop was made in 2009 shortly after the spate of laptop and phone battery fires. That's unfortunate because the entire design philosophy of Li-ion batteries changed about then. Prior to the fires, the manufacturers put the charging circuitry in the battery, under the theory that you could match the charging circuitry to the battery chemistry, so it would always charge the battery perfectly.
Unfortunately, that led to a flood of cheap knockoff batteries from China which would fit in the laptop or phone, but had poorly designed charging circuitry. People would balk at the $150 price of a replacement battery, order a cheap Chinese knockoff from ebay or Amazon, and plug it into their phone/laptop. The first few charging cycles it would work just fine. But after the battery degraded slightly the simplistic charging model built into the charger would no longer work, and the battery would catch fire or explode.
Consequently, manufacturers moved the charging circuitry into the device. That way you can hook up any battery whose specs are close, and it'll charge safely. Unfortunately since this laptop straddles this transition period, there's no way to tell which type of charging circuitry it has. Thus (3a) may not apply.
If you wish to pursue this, I would highly recommend taking the laptop to a qualified repair shop and see if they can figure out if it's safe to plug "any" battery into it (charging circuitry in laptop), or if you must buy the expensive Durabook battery (charging circuitry in battery).
tl;dr - This battery may not be a simple device where you can just swap parts in and out at will. There's a high probability that doing so will result in a fire or explosion. Consult with an expert repair shop before attempting what you're trying to do.