Question 1: How are the backgrounds-colors in all of these images, so perfectly complementary to the subject-colors? Is digital manipulation involved?
The photographer surely made some adjustments that affect the colors, such as adjusting white balance, saturation, highlights, etc.
I wouldn't describe the background colors in the examples you cited as complementary -- the complement of green is red or magenta (depending on the color model), and the complement of blue is orange or yellow. The colors are certainly different and provide some contrast, but not all of Bertner's photos are like that. Some have similar colors (e.g. a green frog on a green background). I think that Bertner, like any photographer, looks for opportunities to compose an attractive image, and that often involves looking for contrasting background colors.
Question 2: Is the unusual texture of the background natural?
There's nothing unusual about the texture -- it's just heavily blurred. It's actually difficult to avoid that sort of strongly blurred background when you're shooting macro, where the camera is very close to the subject.
Context for Question 2: Most macro lenses peak at f2.8, I've never obtained such a beautiful and smooth background even at this aperture.
You don't even need to go as wide open as f/2.8 to get that kind of blur. Here's an example I created in 60 seconds:
The subject here is a Lego minifigure standing on the back of my sofa. If you've ever seen a mini figure, you'll know that the photo covers a vertical distance of maybe 1.5". This was shot using a 100mm f/2.8 macro lens at f/5.6, and the distance from camera to subject was around 6-8" The red background is a textured red blanket that happened to be laying on the sofa. Here's another shot of the same scene:
I shot this one from further back, maybe 18", and at f/32, so that you get more depth of field. As you can see, you don't need to shoot at an enormous aperture to get a lot of background blur if the subject is close to the camera.
Question 3: Is it possible that gels are involve in the creation of such complementary colors?
It's certainly possible, maybe not even unlikely, but I don't think it's necessary to assume that -- the colors don't look unnatural to me.