Although it's probably still to early to be making calls, here's what occurs to me.

[hider=The Technocratic Union]-The Technocratic UnionThe Stoor probably treat the Technocratic Union with some degree of amusement, as though the entire collective is a big, puerile joke.

They probably wouldn't even deign to acknowledge the Sextanis Syndicate as a political entity; Stoor excursion forces would probably regularly violate their borders and seize every other fifth vessel they came across just because. They would probably not take any offers or attempts at diplomacy seriously, and spend the majority of such efforts making veiled threats. To them, the Sextanis Syndicate is just a collection of even /more/ humans, only half of these losers are worthless, complacent layabouts calling themselves 'transhumanists.'

They probably treat the Corvexians with no small amount of condescension, viewing the whole lot of them as a collection of vagrant slaves that haven't been processed yet. They'd perceive the lack of a strong government as tacit permission to exploit the populace. The Stoor would probably skip the saber rattling and attempt to break into the Corvexian marketplace, starting up a slave trade if there isn't already one and attempting to weaken the any semblance of Unity amongt the Alliance through deliberate disruption of the market. Every few weeks they would probably 'vanish' particular shipments or vessels just to flex their muscles.

The GARDS are at about the same level as the natal Stoor that are repurposed for use as autonomous door openers installed in the walls. The Stoor would probably just go about pretending they don't exist.[/hider]
[hider=The Hau Union]The Stoor would perceive the Hau as civilized and strong - they view the use of a Republic Monarchy and a strong center of government to be a source of strength and stability, and they'd view the general harmony amongst the diverse species to be a sign of cohesive unity - in other words, too inconvenient to try to subvert through indirect means. Their excursion forces would likely stay outside of the Union's borders as long as they aren't antagonized by patrol groups or fleet units, and they would probably limit seizure of vessels to those that stray too far outside of Union territory. The Stoor would likely go out of their way to establish direct diplomatic proceedings in order to hammer out trade negotiations and exchange deals - this arrangement would likely be a backroom deal with the Monarch and select members of the Senate, more of a favors for favors situation than anything else. Publicly, the Stoor diplomats would probably record all diplomatic proceedings on memory modules and then sell them as comic memories of some asshole Stoor jerking everybody else around.[/hider]
[hider=The Graal Hegemony]The Stoor have a very simple division of power, and they generally see their home system - the only part of their large territory worth attacking - as unassailable due to the by all standards massive number of large fleets they have parked there doing nothing. They would probably regularly taunt and attempt to provoke the Hegemony as a whole with frequent border raids and skirmishes. Behind the scenes, diplomats would probably try to set up a sort of Orwellian tripod deal with the Hegemony similar to the arrangement seen in 1984, promising to 'wage war' with the Graal in such a way as to help reinforce the Hegemony's control over surrounding systems.[/hider]
[hider=The Human Sphere]The Stoor wouldn't even acknowledge the sphere as existing. They wouldn't do the whole casual-border-violation-because or seize any of their vessels or try to start any shit in the diplomatic sense, because to them, the Sphere just isn't there. 

The only likely avenue of contact between the two will probably be through a third party, or as the Stoor gradually expand their own borders into Sphere space and start salting planets.[/hider]

What do you all think?

Edit: I can't make any real analytic decisions about the Virus without being told more about it - existing information is too vague.