all the realtor websites will have the available listings displayed, so you can check if they have any cabins or maybe ask one to refer you to somebody who does. hopefully that helps some.
References :

Phantom Ranch at the bottom of the Grand Canyon takes reservations 13 months in advance and all spaces are usually filled quickly (both dorms and cabins). If you are planning this for anytime before maybe December 2008, than odds are that it is going to be too late to get reservations (although sometimes you can get lucky with last minute cancellations). Phantom Ranch (like the lodges on top) are ran by Xanterra and you can make reservations online (see link below).

The other way to spend the night at the bottom of the canyon is to backpack down and stay in the Bright Angel campground which is near Phantom Ranch at the bottom of the canyon. This is the place that requires a backcountry permit mentioned by the poster above. They open these permits for reservations on the first of the month, 4 months (not 90 days) in advance. In other words, on January 1 (or more likely Jan 2) they will start filling permits for April based on the requests they have received. Permit requests must be mailed or faxed in. They usually save a few for last minute walk-ins (good for the following day), but you have to be at the office at 8AM and can’t count on it. See link below.

Regardless of if you stay at Phantom Ranch or backpack down (carrying all your own gear), as the the poster above mentions it is a tough hike out (7-10 miles and a 5000 foot climb). If you are a strong and experienced hiker and go up South Kaibab, you might make it out in under 4 hours. A typical hiker going up the Bright Angel trail might take 6-8 hours or more. I have frequently hiked out of the canyon and then driven back down to Phoenix (3.5 hour drive) on the same day, but it makes for a long day. Lake Mead would be an even further drive (5 hours plus) and you would be hard pressed to get there in time to get a boat.

I would recommend just going to Grand Canyon and spending two nights on the rim. This would allow you to relax a bit, see the historic buildings and take in the overlooks. You could even do some short day hikes down into the canyon. The grand historic 1905 El Tovar hotel has honeymoon written all over it (again though… need to get reservations from Xanterra as far in advance as possible).

Grand Canyon is a great place and would certainly make for a memorable part of your trip, but it sounds like you are trying to pack way too much stuff into a short period of time. You obviously want to have a great time on your honeymoon, but remember the most important thing is the two of you being together, so leave plenty of time to relax and enjoy that.

Lastly, I would strongly recommend that you NOT go to Grand Canyon West (aka: the West Rim) where the new glass Skywalk is located. This is a money-making venture on the Hualapai Indian Reservation and is NOT part of the National Park or anywhere near it. The canyon at GC West is not as big, deep or scenic as in the park and you are forced to stay on packaged tour (no wandering or hiking on your own) that typically costs $80 per person. Many of the promotions by GC West (and the Vegas tour operators that go to it) are very misleading and many visitors come away feeling ripped off (just read the TripAdvisor.com reviews for it). I have been there and found it to be way over-hyped, way over-priced and poorly ran.
References :
Arizona resident and frequent Grand Canyon visitor and hiker. Have stayed both at Phantom Ranch and backpacked to Bright Angel camp numerous times (as well as stayed at all the rim-side lodges).

Xanterra (Phantom Ranch and lodge reservations):
http://www.grandcanyonlodges.com/

Grand Canyon backcountry permit info:
http://www.nps.gov/grca/planyourvisit/backcountry-permit.htm