1- I'm pretty sure the Tev part of the GTVA is still in one piece thanks to assassinations and propaganda. Between the constant threat of the Shivans, friends dying against other Terrans in Sol, the economy at an every-time low, and Zods next door with a flourishing economy, the civilian Tev population is a powder keg ready to blow at any time.
2- Let's be honest : although the Tevs should of course try to avoid unnecessarily losses, they are not, contrary to the UEF, on the verge of logistical collapse, and, like Simms or I don't remember who said, they "can bleed corvettes left and right". Steele has shown with Darkest Hour that he is ready and willing to sacrifice a few ships to achieve important goals. If he gets an opening to wipe a Solaris out, he'll take it.
3- Yup. Like said before, the UEF is on the verge of logistical collapse. Anything that can aggravate that situation is welcome for the Tevs.
4- As stated several times in that thread or somewhere else, they can't just dispatch another destroyer to Sol. There is a logistical reason, because a destroyer battlegroup can't just pop in and get busy without supplies, ammo, fuel and whatnot, but if that was the only reason, I'm sure they could have found another Anemoi to take care of that. The real reason is : because Shivans can return any time, any place without warning, they need to ensure a sufficient military presence in their territory. Not only because of the threat alone, but to reassure the population ; after all, the GTVA's political strategy has always been to build a strong armada to defend their citizens.
They also can't afford to send a negative message by showing they can't handle a puny, crumbling fleet of pacifist and need to send several thousands more soldiers into the fray. Four destroyers in Sol is already a pretty large number, it's 1/6 of their fleet, and they're their best elements.
5- Steele has shown to be just as good in term of PR than he is on the battlefield, first by "letting" Severanti make his huge Luna mistake and get removed from command, and then by convincing the Zods to lend logistical support for a war they were originally trying to stay as clear from as possible. If he can pull the good strings to convince the Zods to lend more support without straining the relations between the two species, he will.
Another point being that the Zod admiral of the Shepseskaf was very quick and eager to "fall for it" and believe word-for-word what Steele spoon-fed him. I don't think all Zods share the opinion of the Emperor about staying away from the war. That Zod admiral might be of quite a belligerent nature, and try to find good occasions to beam some Feds.
6- That is true. The GTVA can't cover everything with propaganda, and they need to avoid worsening their image any further. Not to mention that officially, the GTVA considers all Terrans in Sol to be GTVA citizens under the oppression of the UEF, which means that killing civilians equals killing GTVA citizens.