Is freight shipped on certified air carriers?
I am not exactly clear on what you mean by a certified route air carrier, but as to fuel efficiency, or overall efficiency there are several things going on behind the scenes. You are already aware of the major carriers of parcels (FedEx,UPS) then there were Airborne Express, DHL, Purolator,Burlington, etc... these were actual companies with their own leased/purchased aircraft that flew their own routes etc. There were smaller companies such as Emery or Fine Air that would work under another companies FAA approved certificate and have their crews provided by someone else (Rosenbalm). You see this a lot in freight heavy areas such as around Miami where material comes into the country and is dispersed from that point. Then there are freight forwarders that work simply off of discounted rate from the main carriers themselves.
PILOT Airfreight for example, would contract to ship so much per month/quarter/year... whatever... and they would in turn get a discount from the carrier...so the little leased van would drive out to your place and pickup your boxes, and then take them to the UPS terminal where they would be shipped to a holding point and a Pilot van there would go get it and deliver it. By doing this, the main carriers keep their airplanes full (because discounted freight would be bumped in favor of full price freight) and while not as profitable as carrying through their mainline, they helped defer costs which is what it is all about in this business.
The freight business is extremely competitive, and contracts come and go with the wind. Not delivering on time, or misconnecting freight, shipping to the wrong city, or worst of all, losing the freight... and I have seen contracts change twice in 10 days. Pilots are required to fly in conditions that push the limits of the rules, and quite commonly push well past the limits, but when they said "It has to be there positively" it means that you know the law of averages is going to put you in a bad position eventually. The smaller the company, the more dependent it becomes on keeping a customer happy, as the loss of a single contract could be enough to close down the whole operation.
Your question about significant amount shipped using general aviation, I take that to mean Part 135 operators which fly non-scheduled (charter) trips as needed. Significant amounts... as in tons, no.. they can't begin to compete with the large carriers, but they provide for a niche market the "hot shot" trips that are vital to industry. This applies often with the auto industry and you will see parts moved from Mexico thru the USA to Canada for assembly lines that must be kept running. Sometimes it is the size of the door of the airplane that can accomodate certain objects that will determine the company that can provide the service. Still there are companies that operate large turboprop aircraft such as ZANTOP that move large amounts of freight each week but are in a more selective market (auto parts) than the main carriers.
Freight and mail is also moved on commerical passenger flights, but their rates are usually higher than dedicated freighters, but it is a method of getting things moved. You can even send something "counter to counter" on the airlines (or could previously may not be allowed now) but the cost was extremely high.