Stronger With Age: Averted. The eldrae, and various other immortal species and immortagen-users, don’t generally get stronger with age, short of grafting in more advanced biomods and other enhancements as technology marches on. (On the other hand, they do generally get more skilled at applying that strength, and remarkably more cunning, devious, and involuted – but like it says, that’s not this trope.)
Precious Puppies: As a guide to just how straight the eldrae play this particular trope, even where their giant wardogs are concerned, it should be noted that under the list of special cases that is the Ungentlemanly Behavior Act (47, As Subsequently Revised), the consequences of literally kicking the dog are legally recognized as a type of suicide in every single Imperial jurisdiction.
(While there are some people who in private would admit that this constitutes Disproportionate Retribution even by the Empire’s, ah, generous standards for such, there is absolutely no-one willing to court the plummeting reputation score that would attach to anyone who suggested removing this provision, the puppy-hating bastard. It’s a mélith thing; dogs give essentially infinite loyalty, so that’s what they get in return.)
((And, of course, this is also pretty much what started people off on the Immortagens For Everyone crusade in the first place…))
No Immortal Inertia: Averted. Biology does not work that way. If you stop taking – or somehow take away – the immortagens that made your cells all youthful, well, your cells are still youthful. You’ve got a full natural lifespan to go find some more.
Unobtainium: Of many kinds. Sophisticated materials science is one of the major areas of advancement in this particular universe. Of particular note: deuterium slush, metastable metallic hydrogen, helium-3 and antimatter (more specifically, antideuterium slush) for power, room-temperature superconductors, sapphiroids (the trade name for the high-grade kind is Adamant™ – not adamantium, because it’s not an element; after all, transparent aluminum has been used, even if accurate), carbon nanotubes, highly refractory cerametals and metallic glasses, muon metals, strangelets, raw tangle – oh, and fun nonbaryonic things like exotic matter (you make stargate frames out of it), gluonic string (held together by the strong force, thus with the best tensile strength available), and so on and so forth. Less elementally, various nanofluids with fascinatingly exotic behavior, nanotech composites, and smart and biomimetic materials (living metal, nanowell-bearing programmable matter, etc.), computronium (okay, that’s not an element either, but…), and again, and so forth.
And medically speaking, of course, immortagens.
Mayfly December Romance: Mostly averted; in two different ways. The happy one is that immortagens are now available for just about every species, thus avoiding the tragedy of death. The less happy one is that… well, while one may well be able to love someone who is doomed to die, inevitably, it’s much harder to go on loving someone who is determined to mortal themselves to death when the price of immortality is a couple of thousand local currency units and a month or so of post-injection fever and chills.
People who prefer suicide to such a relatively low cost of staying with you are not, I submit, very lovable.
Longevity Treatment / Immortality Inducer: Immortagens, which are Exactly What It Says On The Tin, along with mind-state backups. (Previously anagathics, too, but technology marches on, etc., and why settle for a few extra centuries when you can aim for eternity?)
The Empire, or groups within it rather, works hard on selling these to absolutely everyone, on the entirely reasonable grounds that anyone who wants to die and/or doesn’t want to not die is obviously nuts, and anyone who advocates for death in general is even crazier than that. (See also: Immortality Immorality.)