If you want to run fast, you need somewhat structured training. Normal bicycle commuting is certainly beneficial to ones health, but without structuring it it will be often too short fort endurance training or too slow for speed training. If you want to be fast (on the bike, but there is some limited overlap) you need to train shorter fast sections - like sprints and short time trials. If you want to have endurance for long activities, you need to train long rides. Those can be your commutes, but you would have to significantly extend it.
The overlap to running is there but is not perfect, the sports use different muscles to some extent (as I reminded myself painfully when returning to running very recently), but at least you use the same lungs and the same heart - but also many common muscles. You will certainly also need to train running specifically and the same logic applies. Short fast intervals for speed, long runs for endurance, but there are many detailed advices on running-specific sites.
Discussing your diet is really a completely different topic and cannot be done here. Look elsewhere, but normal healthy diet with balanced income of basic nutrients should be the start.