I had my own ideas that I was working on, one of them was Red Mage. I decided to fit it into the class/job system we currently have. You can view the PDF here, https://db.tt/ZA9CVNM7 . And while you made them a DPS, I went in a different direction and made them Tanks, to not only give us more diversity, but to add more Tanks to the roster.
I did get an FC mate to look at Red Mage and they said most of what I had was good, though they felt my chainspell trait fwlt a little underwelming. They had a suggestion like yours, that there would be a proc after using a melee attack, but I pointed out that the point of chainspell is to literally "chain" the spells together, not switch back and forth. I offered a suggestion about a buff you get for like 10s, but I'll make changes once I have more feedback.
Most of your move sets seem good, as I had similar thoughts. I like the idea of Red Philosophy so that they only have to worry about one stat for damage dealing. My lore for them was that they engraved runes on their rapiers which allows them to channel the aether in their blades to perform magic and do high damage. As for you other abilities, I gave the, -a spells as you put them, different elemental names, I have Dia though it's a "Holy"-ish spell for them. As for a DoT I gave them poison, as we really don't have access to that spell yet. I didn't give them cure, since it would seem pointless in FFXIV. As for the spell spikes I made a trait that adds debuffs to the already existing elemental spells. I have the En- spell though it's only one, as only one would be needed. And I have Phalanx, though I kept it as a defensive ability, not made it an attack. As for Dualcasting while I thought about the "cast 2 spells in succession" aspect, I thought that might be a little hard in an MMO, so I made it hit for double potency, with a chance, to make the next spell be instant with no MP, though I might change this after some feedback.
Also if you follow this link, http://tinyurl.com/ff14arrjobs you can see the other jobs I worked on! Dancer, Salve-Maker, and Geomancer. They are speculations so far, but with some feedback and commentary I might be convinced to tweek them a bit. And so far from what I've heard they all seem pretty solid so far, yay.
I gave my Red Mage rapiers, even some attack names are fencing moves too.