Congrats on continuing your commute into the winter! I do agree that biking is therapeutic, as long as car drivers don't do anything too stupid. As to your questions...
1. Do thicker 700c rims exist?
I assume you mean wider? 29 inch mountain bike rims are the same diameter as 700c rims, which is what most hybrid bikes use. So you could get a wheelset that was made for a 29er, which would have wider rims. That said, it is unlikely that you will actually need wider rims. I took a quick glance, and it looks like this bike came with 35mm wide tires. You probably won't be able to fit a lot wider of tire in the frame (the tire will hit the frame), so there isn't a need to get wider rims. However, if you plan to purchase a cheap wheelset, look for one that is labeled "hybrid" or "touring"; these use wider rims than a road bike wheelset, but are narrower than 29er rims.
2. Im not sure what 700x28c etc and 700x38c etc. Can someone give me simple layman's terms?
These are both tire sizes. The "700" portion refers to the size (diameter) of rim that the tires will fit, 700c. The "28" and "38" refer to the width of the tire in millimeters. There are differing opinions on what width is the best, but are based on the type of winter you have (ice, snow, rain, all of the above) and what type of streets/paths you ride. Instead of giving my opinion on this, just do some reading on other threads, and you will probably get an idea of what is best for the surfaces you will be riding.
2. Is it easy to put the gear on the rear rim hard?
Do you have some basic mechanical skills? If so, there is no reason you can't put it on. However, you will need several tools, one or two which are bike-specific; a cassette tool and (possibly) a chain whip.
3. Upon doing research I came across a thread where someone mentioned that sealed hubs are needed for winter?
Most modern hubs come with some level of sealing, though many mountain hubs are better than road hubs. More important is to overhaul any hubs at the end of winter (replace the bearings and put in new grease). Speaking of hubs, it is important that you determine what width of rear hub your frame uses (130mm is road standard, 135mm mountain) if you buy a separate wheelset for winter use.
It seems like you are interested in doing your own work, so do some reading, hang out at a bike collective, or find a friend who can show you the basics. If reading, there are a couple websites that I would recommend. They are
http://www.sheldonbrown.com/ and
http://www.parktool.com/blog/repair-help.
Good luck!