I think you guys have covered a lot of it here already. One of the principal reasons is the lack of small powerplants that are useable for this application. Cost is also a definite consideration and if the piston engine can do what you want, then it is generally the way most people go.
The smallest practical free turbine engine in common use is the Allison 250 series. Although it was principally designed for helicopter applications (Jet-Ranger, MD 500 etc) the availability of a compact, small and light engine developing lots of power attracted considerable attention as a replacement for various piston engine applications that simply would not physically permit a larger piston engine to be installed.
Joe Soloy figured he could install these engines in the Hiller 12 and Bell 47 helicopters, making some pretty potent machines and he created an entire business around it! Later on he realised that the same potential existed for putting more hp into fixed-wing aircraft that did not physically have the space for a larger piston engine (generally because the next size up was a radial)! As a result Soloy created and certified the 250 powered turbine Cessna 206 and 207. There have been many other mods including Cessna 210, Beech Bonanza's, etc.
Larger aircraft utilized the PT-6 and Garrett 331, generally as replacements for radial engines. Pretty much all the good radial powered aircraft ended up with a turbine engine at some time (Beaver, Otter, Beech 18, Goose, DC-3!, etc..) but perhaps we are at the limits of General Aviation now.
The penalty with these engines is the balance between performance and fuel consumption. Often some mitigation is achieved from the increased speed, allowing an equal (or improved) effective range. The ultimate decision is whether you can gain a really useful performance increase in return for the added operating cost and complexity (plus the cost of engineering and certifying the change).
Noise is not a significant issue with these engines, as they are very quiet to start with and because the engine inlets are buried deep within the cowl, any compressor noise is well muffled. Coupled to low rpm props, these aircraft are some of the quietest around, especially compared to their piston engined variants! We were once air testing a Caravan amphibian, landing in a very popular lake on the outskirts of a big city. Amongst the water-skiers and sail boats, no-one paid the slightest attention to the aircraft, as in many cases they never even heard it.
In the search for pretty low horsepower turbine power, the APU's are really all thats out there. In their normal design configuration they are designed to produce the maximum amount of power from a minimum of weight and complexity. This is achieved (typically) with a single stage centrifugal compressor - hence that ear splitting racket they make. They are also designed for 2 running speeds, On and Off! Smaller and smaller jet engines are being produced today, and there is always the possibility that someone will produce a turboprop variant at some point. I think that the diesel engines that are currently nearing certification are the real way forward for General Aviation and will end up as a retrofit replacement for many existing aircraft.
I have operated a Soloy 206 and a Garrett and Pratt powered Beaver (all on amphibs) - so how do they compare to their piston variants (in my particular application)? The 206 was WAY better than any piston 206 (including the 550 conversion), but a piston Beaver was a better machine all round - so it got replaced. The turbine Beavers really perform well, but the piston one was fine (plus you gotta love the sound of the 985)! The turbine Otter was a great machine but the Caravan beat it hands down for speed and range (but loses on ability to get out of the water) - so its all in the reckoning!
The problem for any upgrade to an aircraft is considering the cost of the modification, you automatically end up comparing performance with the next size aircraft up. Sometimes though there are limiting performance factors that require the smaller aircraft - and boy, I've seen some beautiful machines out there!
As far as aerobatics - not sure about the Allison (don't think I'd trust it for that), but the PT-6 is in the single engine military trainers, so can't see it being a problem - if you've got the money that is!!!!
Anyone interested in more info;
Soloy conversions - check out that 700 hp mod - now that would make a great 206 amphib!
Some motor and application specs.
Soloy 207 jump ship performance!