Is it impermissible and considered bad manners to face one's feet toward the Qibla when sitting in the mosque, or is it a cultural tradition or a newly invented addition?

There is no impediment to extending one's legs towards the Qibla, although some scholars consider it subtly disliked (makrooh tanzeehi) to extend one's legs towards the Kaaba if a person is close to it. In all other circumstances, there is no harm or objection to it.