After going through Reggie's house, documenting all the items that he owned that I also had in my catalog, comparing the prices and doing the math, I've found out that ordering an item through the Happy Home Showcase is 1.2x more expensive.
Here are a few of the items for comparison:
Item Catalog Price HHS Price
Modern Wood Stool 1,480 1,776
Dartboard 1,520 1,824
Exotic Screen 2,250 2,700
In addition, you can only order up to 5 items in a single day. I was able to find additional sources confirming this number here and here, though it's still not stated on Nintendo's official website (which is par for the course for a lot of this game).
So if you want to order something, you'll have to pay 20% more bells, but it may be worth it for those items that you really want and/or haven't been able to find yet. Standard limitations on rare items apply here, so anything you can't order normally through your own Catalog, like the Nintendo set, Gyroids, HHA rewards, etc., you also cannot order through the HHS.
There is indeed a certain order in which you receive the items, up to a point. Afterwards, you receive one of the previously awarded items randomly. The order is as follows:
First time: Nothing
Second time: Balloon
Third time: Balloon
Fourth time: Pinwheel
Fifth time: Bunny Balloon
Sixth time: Bubble Wand
Seventh time: Heart Balloon
Eighth time: Ice Cream
Ninth time: Soft-serve Ice Cream
Tenth time: Party Blower
More than ten times: Random item listed above
The color of the items you receive is random. You also can only Streetpass the same person every 8 hours to receive these gifts.
Additional Sources
Best Answer
The oldest house (that has not been marked as a Favourite) will be removed and replaced with the new house. You can only favourite up to 16 houses.
Source: http://animalcrossing.wikia.com/wiki/Happy_Home_Showcase