It is not rare for a betta to change colors. As long as he is eating and swimming normally I would not worry. It seems like you are taking very good care of him
Is the tank heated? If not, then adding a heater would help Lupin feel more comfortable since bettas are used to water temperatures in the 78-82F range. Another option would be to lower the flow of the filter so that water current in the tank isn't too strong. The small daily partial water...
It's possible to buy just the tank itself and get a separate filter. I'd go with that route since the filters that often come as part of a tank kit are not the best.
Yes, that tank should be fine. Bettas' natural habitat are shallow rice paddies so this would actually be ideal (at least compared to the other extreme of having an relatively deep tank)