When changing aperture size does focal length also have to change to keep angle of view the same?
No, at least not with film cameras. I dunno much about the new digital stuff. Aperture refers to the size of the opening in the shutter, which, combined with the shutter speed, determines the amount of light that reaches the film. The field of view is determined by the focal length of the lens, and remains constant in fixed focal length lenses. It is adjustable in zoom lenses.
Short answer: No.
The angle of view is defined by the focal length and the size of the image plane. To a good first approximation this is independent of the aperture. Curiously if you take into account the physical reality of a lens system, the issue you can run into is vignetting. Here, when you have a very large aperture you can find that the coverage of the image plane is not even, and towards the edges there is incomplete coverage - at the edges you can’t see the entire circle of the aperture because bits of the lens structure get in the way, or internal lenses are a bit smaller than needed, and thus not as much light reaches the image plane at the edge as the middle. However the image scale doesn’t change, and that is really the core answer to the question. You just get slightly darker edges to the picture.
You will also run into edge darkening trouble when the angle of light incident on the image plane is significantly worse than 90 degrees. For very wide angle lenses these issues can be a real problem, but for the majority of lenses it isn’t. The simple answer if there is vignetting, is to reduce the aperture. I have one lens (a perspective control lens) that has markings on it to show the extent of vignetting and maximum apertures, but it is a very extreme example. Compact optics on digital cameras may also suffer from vignetting, however because the camera has full knowledge of the optics it can compensate. High end RAW image processing system can also perform such compensation if the image contains enough meta-data to describe the lens system.
ETA, ninja’d as usual. 
Angle of view should only depend on focal length and crop factor (sensor size), which together can be expressed as 35 mm equivalent focal length.
I wonder if, for common focal lengths, the light can be considered uniform across the sensor? Because for telescopes the decrease in light away from the axis has to be calibrated for.
My question came from the thought that aperture should let light from (at least slightly) larger angle than for example pinhole camera (see: http://i.imgur.com/Ga16P.jpg) but I guess I’m misunderstanding something…
The pinhole camera substitutes a pinhole for the lens. I believe the term you may be searching for is “cricle of confusion”. The image for a pinhole camera can be no sharper than the size of the pinhole (ignoring diffusion effects). Each element of the picture - a spot n a leaf, a spot on a flower, etc. - will shine through the pinhole and create a circle on the back wall of the camera the size of the pinhole.
With a lens, a certain distance is in focus. anything before or after that distance from the camera lens is out of focus.
Consider a point source of light. If the light is right at the focus distance, it will also be a point on the sensor/film. If the light is closer to the lens, it should focus as an image of the point closer to the lens than the sensor. Think of it as rays from all the lense focussing like a cone into that point. Since nothing stops it, that light then goes through that point and keep spreading - once it reaches the sensor, the image is a circle instead of a point (assuming circular lense). Make the aperture smaller and the cone is less of an angle spread. The circle of confusion for “out of focus” is smaller.
Similar for a point source behind the focus point of the camera - the focus of the image would be behind the sensor, the image on the sensor is the circle because the cone of focus is intercepted before it can focus.
By reducing the aperture you make the “spread” of the out-of-focus objects smaller. If the circle of confusion is around a pixel, the object is as good as in focus - so a smaller aperture means that more of the image seems to be in focus - a greater “depth of field”.
Most cheap point and shoot cameras have focus and paerture preset so that almost everything from 6 feet to inifinity is in focus, no need to move the lens.
It does mostly depend on the focal length, so the drop towards the edges is most significant for wide-angle lenses. However, with digital cameras, the camera can compensate for the effect if the properties of the lens have been programmed into it – and even if the camera can’t compensate, a program like Photoshop can too.
I mean, the telescope I used had a very long focal length - couldn’t get a shot of the whole moon - and still needed calibration, so I’m wondering is it:
-telescopes must be really precise
-digital cameras do it automatically, with Bayer interpolation.
Come to think of it, film cameras are fine, so it sounds like the first case.