Regarding the bottom bracket, a picture would be helpful to allow us to determine what it's got.

As for the sprocket (not the chainwheel, that term only refers to front sprockets), there are a number of issues that could be taking place. Firstly, the pawls in the freewheel mechanism could be gummed up with old grease. This isn't uncommon, and can be fixed by flushing or cleaning out the freewheel mechanism.

Secondly, the pawls or ratchet in the freewheel mechanism could be worn out, or the pawl springs could have failed. Repairing this will require new parts, and depending on the model of hub, replacement parts may or may not be available.

A third possibility is that the sprocket is worn and that the chain is skipping on it as a result, which is more likely to be an issue if you've recently replaced the chain with a new one. Again, if the sprocket is worn, it'll need replacing, which may mean you need a new sprocket, a new freewheel or possibly an entire replacement driver assembly depending on the type of hub you have. What model rear hub is it?