A hot topic!
Well, there are some things you can do, and others you can't. The good news is that AGAs will work seamlessly with some renewable sources and integrate well with a 'renewable home'.
It isn't possible though to run an AGA on 'biomass' fuel whether it's pellet form, chipped or plain logs - wood just doesn't have the right 'calorific consistency' or energy density to work properly with the AGA.
(The solid fuel AGA [discontinued many years ago] needed to burn specific types of coal or coke; this was the only fuel that gave the correct amount of heat and the long burn times between fuelling for convenient running. The solid fuel AGA needs riddling and filling about twice per day, this would be more frequent burning wood.)
Biodiesel isn't compatible with the oil AGA's 'vapourising' burner (which is extremely sensitive to any impurities, never mind a different grade of fuel!) - it would cause incorrect combustion and greatly reduced service intervals so just isn't suitable.
Yes, perfectly compatible with any electric AGA.
The array of solar panels are connected to an inverter which is in turn connected to your home's supply and the 'mains grid'.
When you're generating power any excess you make will 'feed in' to the grid, and when you need more than you're making the difference is drawn back from the grid. (The mains grid effectively acting like a huge battery.)
The AGA still connects to the house supply as normal (the panels don't connect to the cooker) so always maintains its temperature and cooking performance whatever's happening to your 'micro generator-set'; it'll use power from either your local generation if it can, and draw extra from the grid should it need to.
Yes, again, both are suitable to generate power for an electric AGA.
Domestic systems usually connect to the grid in the same way as solar 'PV' so the AGA is connected to the house in the conventional manner, and works seamlessly with whichever source is making the electricity at any given point.
Not normally practical in a domestic situation.
Biogas is produced as a byproduct of 'anaerobic decomposition' of organic waste, often from landfill sites or farms.
Composed of several constituents (primarily methane) the gas usually needs treatment to make it suitable for burning either to produce heat directly, heat to power a turbine (then generating electricity), or as fuel for a biogas engine which generates electricity. The electrical power can then easily be used to feed an electric AGA.
The weather isn't consistent enough to design an AGA round one particular renewable energy source; we'd recommend choosing the right AGA for you, your lifestyle and your house's heat requirements and the right grid-connected renewable source for your circumstances. With the energy grid acting as a giant battery you'll know your AGA and renewables are working to a harmonious solution.