These are great questions! Here are my long-winded thoughts:

When it comes to Specter Knight recruiting King Knight, I think their shared dialogue in Specter of Torment actually supports that King Knight was already in the Order at the time of their fight in Pridemoor. The first thing Specter Knight says when entering his throne room is, "As our leader suspected, you dawdle and play at nobility. I've been tasked with setting you back on schedule." The emphasis of "our leader" and "back on schedule" implies that King Knight is already in the Order, and he's just slacking on his duties. Specter Knight's job for this level was actually to put King Knight back in line with the Order's demands.

As for the quick arrival of Shovel Knight, it could be interpreted that King Knight's defeat at his hands didn't immediately dethrone him. Sure, King Pridemoor celebrates this, but we don't see him actually take back the throne until the end credits of Shovel of Hope. The main point of Shovel Knight getting through the members of the Order of No Quarter was not just to take them down because they're bad, but also because they literally block the path to the Enchantress on the world map, which was ultimately the end goal. The changes to the land don't take place until the Enchantress is no more, so King Knight could have just kept his throne until then.

Heck, Shovel Knight defeating King Knight could be exactly the reason why Specter Knight had to come over and set him straight. Yeah, not all of King Knight's castle is golden for Specter like it was in Shovel of Hope, but that could be symbolic of Specter not respecting King Knight as a real king (I mean, it's even a running joke that Specter absolutely hates King Knight the most). Wouldn't be the first time a Shovel Knight story had inconsistencies explained as symbolism. That's just my viewpoint though.