Using One Hot Encoding VS Label Encoder for Football Match Outcomes
I would expect some ordering, since the expected outcome is probably depending on the team's strengths. However, there may also be a tendency to/away from draws if teams are equally strong. I would guess that the one-hot encoding is the better choice if you have sufficient data to train your model.
Your prediction vector would typically give e.g. [0.1 0.7 0.2] as probabilities, and you select the highest probability as the outcome. Either by hand or in a (final) layer in the model.
It shouldn't. Typical models are dimension agnostic. E.g. a fully connected first layer loses all positional/dimensional bias. For trees similar arguments hold, however, I don't know how well tree models would perform on such sparse data.
The bookies' odds should highly correlate to the expected outcomes, e.g. bookies will likely have odds that reflect relative strength. In such a case, the odds could more or less 'vanish' in the model.
With such sparse data, you should definitely tune your parameters for forest models carefully.




Just some added notes: If you want to improve on your model, you could also include a few calculated parameters, e.g. current rank, win/draw ratio for both teams (in general, or just the respective home/away versions), most recent (home/away) results. Or even travel distance if you have the data.