To me, the fact that it recovers overnight suggests it's getting too much heat or not enough water. But I'm not an expert. When you're growing in dark-colored containers in direct sun, there can be a problem with the soil heating too much. So if that's the case here, you might cover the pot in something that will absorb less heat from the sun: for instance, a white plastic bag, a piece of white cloth, or a sheet of silver mylar (space blanket stuff). If you don't get more answers here, you might consider cross-posting on the Container Gardening forum. I have wild ground cherries, but I've never planted them, let alone indoors. The flowers dropping might be related to whatever is making the plants droop, or could be a pollination issue. According to the California Rare Fruit Growers, "Cape gooseberries are self-pollinated but pollination is enhanced by a gentle shaking of the flowering stems or giving the plants a light spraying with water." Shaking the stems a bit is also recommended for indoor-grown tomatoes.

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Mine volunteer in large numbers also; the winter freeze doesn't seem to hurt their germination. If you start them indoors, I agree with @wayne_5 zone 6a Central Indiana; don't plant them too early. They are fast growing & tend to develop a tap root, so if started indoors along with tomatoes, they may become root bound. You can direct-seed them at the same time as beans, and still get a fairly decent crop... but the best crop will be the volunteers you get the second year.

This Ground Cherry was found growing next to a river. It was probably close to ~2' tall with cherries about a ~3/4" diameter. Which species exactly is it?