You've asked a few questions here.
As for the brightness of Jupiter, Jupiter has an albedo of around 0.5, compared to 0.12 for the moon. This means that Jupiter will reflect roughly 4x the light that the moon does. However, Jupiter is significantly larger than the moon. You've indicated a 30 degree angular diameter for your Jupiter, this being 60x the diameter of the moon, will result in 3600x the angular area of the moon. Combined this means the Jupiter will shine roughly 14,400 times as brightly as the moon. Given that the moon shines at 0.05 - 0.3 lux, a "full" jupiter will shine with 720-4320 lux. When half of Jupiter is illuminated, it will shine with half this lux. When a quarter is illuminated, it will shine with a quarter this lux etc... This is very bright.
You've asked how bright Jupiter will be during the night. The brightness of the Jupiter will be the same everywhere on the planet at once, but the time of day everywhere on the planet will be different. Because the Earth is now tidally locked to Jupiter, there will be a certain region where Jupiter is constantly straight up at all times (in reality, the Jupiter will wobble a little bit depending on how circular and angled the Earth is orbiting, but it won't be much). This point will have a fully illuminated Jupiter during its midnight, and a halfway illuminated jupiter at dawn/dusk. The darkest it will ever get in this region will be when the Sun is being eclipsed, which will be often.
Depending on where Jupiter appears to be in the sky, Jupiter's phase will be offset from the day/night cycle by some constant amount depending on the longitude. The lower Jupiter appears in the sky, the less sunlight will be present during Jupiter's "dark" phase, giving a true "night". Also worth noting, will be that half of the Earth will never see this Jupiter, and thus never be illuminated. This part of the Earth will be akin to the Dark Side of the moon.