How to see all keys for a PrivateTempStore?
No, with the current PrivateTempStore you don't have access to the list of keys. You're welcome to create your own TempStore class if you want different functionality, but I don't see why that is needed here.
The whole point of PrivateTempStore is to stash temporary data for a user, under a given key. If you don't know the key, then you have no knowledge of the type or structure of the data stored there, so in general you won't know how to use it. If your intention is to store a collection of data of a known type (stored in an associative array, for example) so that you can iterate over it later, then store the array itself under a known key, don't store each array element separately. Then you can retrieve the array later from the known key and do a foreach over each element of the array.