DA-20,
You most certainly can get a ATP, single engine land, in a Piper Cub (heck, get your single engine sea while you're at it!), as long as it has the equipment required by the ATP Practical Test Standards for performing the prescribed area's of operation. Those are conviently available for easy reading, at the FAA.GOV website. You don't need a "complex" airplane, as is required for the Commercial Pilot Certificate, and the aircraft doesn't even have to be certified for actual IFR flight (practical tests are usually performed, as a matter of policy, in VFR conditions)
You can earn your ATP -Rotorcraft Helicopter in a Robinson R-22, if you so desire. Sadly, you can no longer get an ATP - Rotorcraft Gyroplane...that would have been a hoot.
In that vien, understand that each grade of certificate (sport, recreational, private, commercial, ATP) was held to a certain level of performance, and that performance was carried over from one grade to the next. If you showed competence in complex aircraft during a practical test for your commercial, single engine land, there was no need to demonstrate it on another practical test.
The ATP is nothing more than another rung on the ladder....another grade of ticket. We don't really view airline ops as the end all, be all as it seems to be in the rest of the world, so smaller equipment works just fine.
Remember, on FAA certs, there are grades (Sport, Recreational, Private, Commerical, ATP), Categories (Airplane, Rotorcraft, Glider, Lighter-than-Air, Powered Lift) and classes (single engine, multi engine, land, sea, helicopter, gyroplane, etc).
Instrument Ratings are applicable to airplanes, helicopters and powered lift. That will be noted on the certificate.
Type ratings only as required. If you do an ATP ride, however, you are generally awarded that type rating, if such a type requires it, since the checkrides are identical.
They can be combined in a pretty much unlimited manner on any one certificate, although not all categories and classes exist for each grade. And remember, they never US pilot certificates never expire, but you can go non-current, which is an easy fix.
Flight Instructor certificates follow the same format, although some classes are combined. For instance, a Flight Instructor certificate may have Airplane Single Engine, which covers both land and sea (you obviously have to have a SES on your pilot certificate). CFI tickets DO expire every two years if not renewed, which is mostly painless process.
There also other types of certificates...Ground Instructor, Flight Engineer, Flight Navigator, Mechanic, etc. They all look identical, except for what's printed on them. Pilot certificates have a blue DOT seal on them, while all the others are black, and if you have a good pass rate, you can qualify for a "gold seal" on your instructor certificate.