Just so you know, I only read up until about the fourth post on the first page. Ignore any reppetition.
10 gallon tanks are not a bargain compared to 20 gallons. In the long run, they cost more money. They hold heat less efficiently (electricity bills), have,generaly, a smaller surface area to volume ratio compared to a 20 long so allow for gas exchange (more aeration needed), they are less stable (fish deaths - think of the cost of replacing them, fish disease - the cost of medications), they hold less so over-stocking happens more easily which leads to the need for more water changes (and everyone knows time is money) and, also due to the tendency to over-stock smaller tanks, greater nutrient build-up which leads to more algae (which leads to the purchase of useless algae-killing products and, more useful, algae scrapers - which, however, do not target the cause and simply treat the symptoms...) and so on... You get the idea though. Seriously, in the long run, if you buy a larger tank now, you won't need to later and your fish will thank (sort of) for it as well.
Now before I deal with the actual question, the difference between tetra swimming and cory swimming is that tetras do it only when they are threatened (or feel they are as the case may be) while cories just seem to enjoy the companionship (actualy, that's anthropomorphising - it's probably more to do with following their tankmates to get to food and things like that).
Back to your original question. You want a tiny tank, so go for tiny fish. Pygmy cories (corydoras pygmeaus), endler's livebearers (arguably poecilia endleri or poecili sp. endler's or even, poecilia reticulata - which, BTW, is the guppy and not actualy the same thing), spotted rasboras (rasbora maculata), sparkling gouramies (trichopsis pumilus) and honey gouramies (colisa chuna or colisa sota). Oh and let's not forget the myers' hatchetfish (carnegiella myersi).
I've listed the scientific names so you can research them (google is wonderful) yourself. Still, I'll tell you some basic stuff for each.
The cories first off - these are relatively hardy fish and very active (as are all cories realy). They are also very social. It follows that they need to be kept as a group. Considering pygmy cories are minute (just about an inch), you can afford to keep them in an adequately-sized group even in a 10 gallon - that size being 4 individuals.
Endler's livebearers are an adorable livebearer that's very closely related to guppies but, IMO, much more interesting. Also, and this isn't just opinion, they aren't as inbred and a lot hardier and way more adaptable. They also only just get to about 1" with females being slightly larger. Only the males are colorful. Having said that, you should stick to just getting males anyway. Besides females often being quite a bit more tricky to come by (as they aren't profitable to stock as they don't sell well), keeping a mixed group of livebearers is begging to be swamped by fry (thus over-stocking and forfeiting all your fish as a result if you can't re-home them quick enough - and they do breed like rabbits, or worse, convicts). If you do keep them as a mixed group, 2 females to every male is the best ratio. Make sure you don't mix endler's with guppies as they'll hybridize (and though guppies are messed up as it is, I would hope endler's remain relatively pure for a while yet).
The spotted rasboras are also tiny fish that only get to about an inch. These need to be kept in a group of 5-6 minnimum. They can also be a little difficult to get hold of but are worth the trouble as they are quite hardy and deffinately one of the better schooling species.
Sparkling gouramies are one of my favourites out of this list (well, gouramies always are) so I highly reccomend them. They are not the hardiest of fish but they also aren't difficult to keep. What they do need is a tank that's got plenty of hiding places and, prefferably, one that's planted. They are best kept as a trio - 1 male with 2 females - but can be difficult to sex at a young age. This isn't too serious of a problem as long as you know all the fish have places to hide and as long as you watch them regularly so you can replace any extra males with females if necessary. Besides sparkling (and yes, they do sparkle), they can also croak when excited. Adorable little things that also get to just over 1"

.
Honey gouramies are my other favourite here (obviously). These are hardy 1.5" fish. Males develop a lovely dark blue-black throat when in breeding condition but females are also lovely, being a delicate shade of honey brown with a slightly darker stripe along the side and a delicate-looking build. Whilst most of the other fish I mentioned occupy the bottom or mid water layers, honey preffer the top. You can keep a single male or get a couple of females (if you don't mind the lack of color). It's also possible to keep a trio if you decorate heavily (a trio, as usual, being 2 females and a male).
The last fish I mentioned was the myers' hatchetfish, one of the tiniest hatchets, if not the smallest. Again, these are a 1" schooling fish that need to be kept in groups. 4-6 individuals is adequate in this size tank and they'll spend all their time near the top. However, this requires that you feed them things that float (or sink slowly) as they won't eat from lower down. They aren't fragile but they aren't hardy either. Don't keep these if you get honey gouramies as they'll be competing for the space at the top of the tank and the gouramies will bully them (as hatchets tend to be prone to being bullied). Make sure you have a tight-fitting hood/lid/cover - hatchets are typicaly suicidal.
Whichever of the above fish you might consider, remember that you can keep a maximum of around 10 individuals (give or take a couple) but you should also remember that some need to be kept in specific size groups (which I have mentioned in all these cases). A lot of them can also be relatively tricky to find - but they are worth it as you can actualy make a community out of them in a way you can't with larger, but also more common, species of fish. Most LFS will order fish in for you or you can order online. Do make sure you fishless cycle the tank beforehand (read the links in my signature), take the time to research each species BEFORE you buy any and take any LFS advice with a grain (or two) of salt.
edit: 'for the record' harlequins are rasboras, not tetras - rasboras are always better schoolers.
I'm hoping you can make combinations of the above yourself but, if you realy can't, here are examples:
1 male honey gourami, 6 spotted rasboras, 4 pygmy cories
OR
3 sparkling gouramies, 4 myers' hatchetfish, 4 endler's livebearers.
OR
4 pygmy cories, 3 sparkling gouramies, 2 female honey gouramies
AND so on... you get the idea?