All the info was lost in the great server crash, but there were threads on there that sorted out the micro/minisprint jungle well. The short version, keeping in mind there is always some track or region that does things differently, or runs a mix not intuitively obvious to those not in the micro or minisprint game is as follows:
The three quarter midget has been around for decades; current rules specify a 750cc motorcycle engine, and the ATQMRA may even mandate a specific series of Suzuki engines; I haven't followed their rules closely enough to be sure on that. In some areas 750cc TQs are raced against against 600 micros to increase field size or make for big fields in large $$$ races. They use an offset chassis with the engine mounted to the driver's left with the cylinders running fore and aft and an adapter to a shaft drive to the cutest little quick change rear you ever saw (with the pumpkin offset dramatically to the left). Any other displacement or chassis configuration, someone is playing fast and loose with the rulebook and it's not a TQ by current standards, but tracks can call a class whatever they want.
Micro sprints: An offset chassis with the engine to the driver's left and the driver's legs passing beside the engine (with a firewall to the left of the driver's legs to protect the driver). Also called a sidewinder chassis. There are several flavors; 250 micros run stock bore and stroke 250 cc motocross engines, the 270cc specimens are bored and stroked 250s. There are often also sportsman/jr classes at tracks that run 125cc motocross engines or 250cc fourstrokes. Then there are the 600s, a larger offset/sidewinder chassis that uses a 600cc motorcycle engine but uses the same layout as it's smaller engined brothers except that there are 4 cylinders running from inboard to outboard. And both the 600s and the smaller micros are are chain drive to a one piece tubular rear axle. Then, to complicate things a bit more in the micro ranks, there is another class, usually called sidewinder sprints, that takes a 600 micro chassis built since 2005 or so and crams a 1000cc motorcycle engine into it. They generally have a rule package that mandates a really stock engine, but with the extra 400cc, that's a bunch more hp than a 600 and a lot more torque. And once in awhile you will still see and old 2 stroke micro (250/270 to 400cc) with the engine on the right instead of the left; they're no longer competitive, but in their day they provided exciting racing.
Mini sprints have an upright chassis like a normal sprint car, with a firewall in front of the driver and the engine mounted crossways in the chassis in front of the firewall. The basic setup used an approximately 1200cc engine years ago, but expense seems to have killed them in most regions where they used to race. In the places where they have survived, they seem to show up as a 1000cc class (often called lightening sprints these days), or a 750cc class, or a 600cc class. The upright configuration uses a chain drive similar to that of an offset chassis micro, but the chain is a bit longer because the engine is in front of the driver, and they generally use 13" wheels/tires instead of the 10 inchers used on micros. We have seen 750 mini sprints run with 600 micros for the same reasons as with the TQs.
Like all forms of racing, micros (and minis and TQs, for that matter) have gotten more expensive over the years, and as with karting, you have to take a look at exactly what your local tracks are actually running, regardless of that they are calling the classes, as you have found out. Based on our experiences here in central PA, if some version of the RaceSaver 305 full sized sprints is offered in your area, it is worth a second look; the rule package makes them awfully cost effective. On the $$ side, there is a lot of overlap in what it costs to run a competetive 600 micro effort for a season and what it costs to run a 305 to the RaceSaver Rule package. That goes double if you already have rollers in your shop - 305s, 358s, 360s and 410s all run the same chassis, so one can progress as little or as much as desire and $$ allow, and you can definitely be competitive with an engine built in one's own garage and put through the proper RaceSaver tech inspections - try that with a 410 or a 360 or even a 358. The only teams I know that are competitive with home garage built 358s don't run on anything bigger than 3/8 of a mile - they can't afford the wear and tear of the larger tracks. The RaceSaver 305s are a very attractive package, especially if there is enough 360/410 competition in your area to generate "hand-me-down" used tires, especially RRs, that the higher roller 410 teams only run for one race, then give away or sell for $20-$40. Spin it up and hit it with a disc grinder or slapper disk and it's good for 4-5 races to half a season on a 305. Take a real close look at the 305s; they are a great training ground and a relatively inexpensive form of sprint car racing.