We were just there last weekend checking in on all of the in-laws. Lots of stuff still closed, at least 2 or 3 organizations still giving out free meals to residents, etc. About 1/3 of my husband's family members had land line phone service working. There is still trash piled up on residential streets. It is being picked up, but there is so much that it will just take a while. There are at least two dump sites where they are taking everything - one by the old Sea-a-Rama and the other near the post office on Broadway. If you have your windows down when driving through many of the residential areas, you get a strong whiff of mildew with a bit of mold thrown in. (People who are there - and don't leave the island - have already gotten used to it and don't smell it anymore.)
Here is a list of what is currently open:
Galveston hotel/restaurant update/. That website is pretty actively managed, so I would suggest you keep checking there for updates as you get closer to your trip. More businesses will be open by December, but as tstone said, service is slow in some places right now and may continue to be so due to fewer employees. You can expect most of Tillman's hotels/restaurants (San Luis, Hilton, Landry's, Saltgrass, Rainforest Cafe, etc.) to be running at about normal speed, though, because he is able to pull staff from other locations if needed and because he doesn't want to lose out on so much potential $$ (I mean he wants to put money back into Galveston's economy

). Moody stuff should be at full power, too. There were work crews at pretty much all of the Seawall hotels during our visit, so a few more of those should be open by late December. The Flagship was in
sad shape long before Ike... Currently the concrete between Seawall and the hotel is pretty much gone.
The Strand was a virtual ghost town except for the people running the hoses to evac. the water from businesses. I do think some Strand area stores/restaurants will be open by your visit, but not all of them will be. Poor Yaga's can't catch a break - I hope they will bounce back.
The Lone Star Bike Rally has been moved to mid-December and I know some places are striving to be open by that time. As you said, the island won't be back to normal by the time of your visit, but it is on its way there. You will have a good variety of hotels and restaurants from which to choose. (If you are staying more than a few days, you might also consider a vacation home rental, especially those managed by individuals. They aren't all open yet, either, but I'm sure most of them could use the money since they will probably be hit harder right away from the loss of revenue. There are a few different websites to find vacation rentals, but you can also find links to them through the main galveston.com website.)
Have a wonderful trip!