No Order of Operations Exists...
These are rules you'll have write. It's unfortunate in d20 that there's no quantified step-by-step order for combat and applying effects a la most trading card games. Were there, we could just say, "Immunity applies during step X, Damage Resolution, after Saving Throws but before Inflicting Effects," or whatever.
...But If You Want My Opinion
Yes, if only so the creature can choose to give up that saving throw.
Immunity shouldn't remove options everyone already has that aren't part of being immune. As everyone has the option of giving up a saving throw before knowing an effect's effect, immunity shouldn't change that.
Therefore when a creature is subject to an effect that requires a saving throw, the creature can choose to either make the saving throw or voluntarily give up (i.e. fail) that saving throw. Then the DM determines if the creature's immune to the effect.
Thus, unless the creature already knows an effect won't affect it (via a successful Spellcraft check, a successful Knowledge check that's revealed the opposition's abilities, or prior experience), the creature attempts the saving throw despite realizing an instant later that it possesses immunity to the effect.
That's because Pathfinder (and its antecedents) is a dangerous place, and what individuals can do varies wildly. A creature is safer if it always attempts saving throws versus affects, even if it thinks it might be immune to such effects because there's usually a random component to identifying effects, and being dumb gets it killed. Even a red dragon--who's immune to fire--will still make a saving throw versus the spell fireball [evoc] (Pathfinder Role-playing Game Core Rules 283) because there's no way to anticipate how a particular caster's fireball is going to differ from any other caster's fireball.
If immunity is checked first there's no opportunity to voluntarily give up the saving throw, and that opportunity should exist even if the creature's immune if for no other reason than to let the DM, when describing the spell's effect, tell the players that the creature appears to give up its saving throw, acting all awesome because the creature suspects he's immune.
If a Precedent's Absolutely Necessary
When spells are printed, they're printed with a Saving Throw first and the Spell Resistance after, so absent another order, those are checked in that order. Linking spell resistance to spell immunity is easy, but making the jump from spell immunity to immunity is harder, but, hey, the words are there.
Let's dive into this spell a bit.
Unseen Servant
"....creates an invisible, mindless, shapeless force that performs simple tasks at your command... The servant springs into existence...on the ground within range. ...you can mentally command the servant to move... and interact with an object. ...can perform simple tasks that a human servant could do to the best of it's ability..."
Now that I've enforced the relevant sections of the text; we can start to answer your questions.
Can an Unseen Servant be interacted with?
- Since the Unseen Servant is not only mindless; possessing no intelligence of it's own and unable to act of its own will; it simply follows the most basic of commands, aided in it's endeavors by the same magic that binds it to you; but is also invisible, as well as shapeless, you would have to have a way of divining the servants location or seeing invisible creatures before you can even attempt to ponder interacting with it. Since it DOES occupy a space, however, this leads me to believe that force applied upon it CAN effect it. It can be pushed out of the way and such.
Can it be commanded to 'hang on' or otherwise interact with another creature?
- It's strength score is 2, that's a -4 . Looking at the relevant entry (do not currently have access to it) we find that a -4 is extensively lower than the average human being. This creature can do simple tasks to the best of it's ability , and it's -4 strength ability probably does not allow for what you are asking.
Can Unseen Servant displace substances?
- Considering that the Unseen Servant is shapeless, as made clear in the first line of text, generally we would have to rule that there is no shape it possesses in which it WOULD displace any kind of matter, liquid or otherwise. Since the spell does not explain HOW this mindless, shapeless magical servant affects the items it does interact with; such as cleaning and folding clothes, we must assume it does so through magical means, NOT with any sort of invisible appendages or form that it possesses.
In closing; this is one of the very few spells that leaves me with holes in it's logic. It is invisible. This one is easy enough. If it DOES possess some sort of form, then you can not see it by mundane means. It is mindless. Possessing of no intelligence or mental acuity, and so it can only follow commands to the best of its abilities, which are very poor and extremely under your average human beings range of capabilities. It is shapeless. This is where the most confusion appears. Possessing of no shape, no substance that comprises it's physical being; whether invisible or not. If it has no shape, yet occupies a space on the ground, how does it interact with the items it must interact with in order to do it's job as per the spell permits? I can't logic this one away. 'Magic' seems to be the only thing holding this together.
Best Answer
Firstly, an unseen servant isn't a creature
The presence of an AC and HP value does not make it a creature, even objects (such as Bigby's Hand, Barrels, and Walls) have AC and HP (and a strength score, in the case of Bigby's Hand). As for the Servant having a Strength score, I suspect it's for determining how much it can lift, drag or carry.
Because the Unseen Servant isn't a creature, most spells that can target a single creature (if the caster can see invisibility), can't legally target it, neither can Area of Effect spells that hits creatures, such as Hypnotic Pattern, affect it.
It can, however, take damage from Area spells that deal at least 1 damage- then it is destroyed.