I really am delighted to hear such praise about my story, thank you!
And while that is a fictitional story, with a different POD and a fair amount of extreme luck on the roman side, I truly believe the empire could have survived with a V century POD.
But to answer your questions:
1) He could but under what circumstances? Eucherius being partly barbarian (1/4) should not be really a problem. Theodosius II was as well. Although in his case his barbarian heritage came from the mother, whose heritage if I remember correctly, only really mattered when the father was unknown (not the case here). Now the question is how does Eucherius come to power?
If Honorius dies at the hand of Stilicho, expect a full blown civil war against both the East (they are not going to stand for the murder of one of their own), like in 423-425, and possibly Africa. In such case he doesn't really stand good chances of surviving. If Honorius dies a natural death and Stilicho can prove that, the only possible legitimate claimants to the throne are Arcadius/Theodosius II in the East, Galla Placidia (she can't rule on her own but she can legitimize the rule of someone else), the Theodosian cousins in Spain or Eucherius.
Now, leaving aside the Spanish cousins (they are too far and probably with the wort claim possible), we are left with three claimants. Depending on the time frame you have in mind, the East might be busy with the whole Tribigild/Gainas/Alaric thing or the issue of the death of Arcadius, leaving them unable to stake their claim. They would intervene at this time directly only if they perceived someone hostile taking power in the West (like a murderer of Honorius for example). Galla Placidia is your best bet but she is unmarried, childless and a woman. No way she can pull an Irene at this point in time. That is where a marriage with Eucherius comes in handy. Eucherius is part of the family (the two are some sort of cousins) and might even get along considering they might have spent time together in the palace before then. Placidia is too young to really have a say, but she might not dislike an arrangement that would turn her into the Pulcheria of the West. The marriage also ensures the support of the army for the current dynasty. At this point, Stilicho can safely get rid of Olympius.
With good diplomacy (which might not be that easy given the mutual mistrust at the time between Milan and Constantinople) and good arguments, Stilicho should be able to get the East to accept the arrangement. The only problem with that is the "cousin marriage" thing between Eucherius and Placidia, something banned a few decades earlier under Theodosius, influenced by Ambrose.
Now the law I think banned only marriages between first cousins (I need to check this one), while Eucherius and Placidia were second cousins. But to be honest the law was quite recent and never really popular (at this time) among the upper levels of society. It was even acceptable at this time for the emperor to bestow exemptions from this law to members of the senatorial aristocracy who asked for it in order to marry close relatives from the same or allied families.
There is then the whole issue of how to handle Alaric's Goths but that is a different problem.
2) As for your second question, yes definitely, even longer than that. But at this point it is quite likely that the two halves would return to be one (keep in mind that the whole concept of two Roman Empires is a modern notion) at some point. Legally speaking this happened in 457/458, 465/467 and on three more occasions (408, 450, 472-4) one of the two emperors showed some sort of claim to handle affairs in the other half. Given the right circumstances, sooner or later one half would join the other (possibly peacefully). The impact of foreign invasion would be altered in this alternate timeline.
I hope I did not forget anything.