Never tried Wasapi, bt with all ASIO based devices I've used on PT I've never had any issues playing music on other programs while PT is open.
Who Me has it right. WASAPI works in exclusive mode with Pro Tools (just). Better off with ASIO driver for internal sound device. For an external device, use the dedicated driver that came with it, if any. Otherwise, ASIO. My preferred ASIO driver is Magix ASIO which can be found on sister site. Will update link if needed.
It's usually a matter of configuration of your ASIO driver. e.g. Charlie Steinberg's multiclient ASIO driver has a special checkbox for this
Doesn't ASIO4ALL also have such an option for exclusive or non exclusive device access? Maybe you could try the Steinberg ASIO driver (open its settings to set it to playback non-exclusively). Link: ftp://ftp.steinberg.net/Download/Hardware/ASIO_multiclient_driver/ Finally, don't blame PT for ASIO4ALL being bad :P
This behaviour is caused by Pro Tools HD itself, using another driver won't change it. Standard (Vanilla) Pro Tools is less picky. Only with a second interface you can play other sources when PT HD is running.