There's no such thing as a dedicated "business visa" for Schengen -- only a uniform short-stay visa, which you can get if you present a coherent story about why you want to visit the area and why you'd want to go back home afterwards. Attending a competition is one such story to tell (and checking the "business" option when the form asks for the overall purpose of your visit is probably the right choice).
Since the competition is clearly your main purpose for going to Europe, you should apply to Portugal for your visa. However it is common and entirely allowed to combine a business purpose with a bit of tourism, especially when you're already traveling halfway around the world. You should provide full details of the entire trip you plan to make in the visa application to Portuguese consulate.
If your expenses are are covered partly by your host in Portugal and partly by yourself, simply state that truthfully. If the application form doesn't allow you to explain the truth in a clear way (note what you can usually check more than one of the options for most questions when several apply), answer its questions as well as you can, and give the full story in a free-form field or a separate enclosure with the application.
