Hi, Ryan, and welcome! I'm not going to be a tremendous amount of help; maybe only a little. But that's never stopped me before.
"Does my myHeritage autosomal data enable me to determine the STR repeats in this way?"
Nope. Not at all, unfortunately. Our inexpensive direct-to-consumer tests use what's called a genotyping microarray chip--something the size of an over-large microscope slide--which has hundreds of thousands of microscopic elements programmed to attract specific points (SNPs, single nucleotide polymorphisms) to them when the prepared DNA solution is washed over them. So those microarrays are looking at about 650,000 specific points, or loci, among your 3.2 billion base pairs, or pairs of nucleic bases.
Short tandem repeats (STR), also known as microsatellites, happen when two or more nucleotides are repeated exactly and the repetitions are immediately adjacent to each other. Our microarray tests don't look at STRs at all; in fact, because of the nature of the chips used, they really can't.
In genealogy we typically only see STRs tested on the Y chromosome where the lack of recombination make them more stable. I'm definitely not as up to speed as I might be about forensics and DNA, but that's where we find autosomal STR testing. Fairly recently expanded from 13, the U.S. CODIS/NDIS databases now use 20 different autosomal STRs. These are super-accurate at identifying individuals, and accurate at identifying paternity.
But the number of STR repeats is highly polymorphic: they're free to change frequently. In fact, there is little risk that a set of tested autosomal STRs will match even a sibling (other than a monozygotic twin) or aunt/uncle. That's why autosomal STRs are great for forensics, but pretty lousy for genealogy. I can't say with certainty, but my opinion is that there would be no way to establish relationship with a great-grandparent based only upon autosomal STRs. Probably not even a grandparent.
Yep; you're correct about the theoretical expected autosomal DNA sharing with your great-grandmother. Those expected averages are derived from something called the Coefficient of Relationship, and are always a good starting baseline to evaluate possible connections. I have a chart here showing those values back as far as (unnecessarily so, but hey) 7g-grandparents. For half relationships--like a half 1st cousin--the values are simply divided in half. The percentage degree of variability increases as the degree of relationship decreases, but these CoR theoretical values serve as a sanity-check baseline.
Folks of a contemporary generation to you from your great-grandmother will be your 3rd cousins. Even our limited genotyping microarray tests can be pretty accurate when dealing with 3rd cousins, so there's some hope in possible verification there. Good news!
I did a quick check to see if there had been any more recent peer-reviewed studies of the Romanovs, preferably using next generation sequencing, which was in its infancy at the time of that 2009 study. Nothing turned up. But in July 2018 The Moscow Times reported that some new testing had been done in which "DNA samples from the living descendants of the Romanov family were used to confirm the authenticity of the remaining bodies." That wouldn't have been STR testing, but whatever research was conducted by Russia's Investigative Committee hasn't been published for public consumption...at least not that I could easily locate. May be worth hunting for, though.