1. Imperial Marines vs. Imperial Star Marines:
I don't think there's any canon definition of this.
I think that both are valid names. Their "true" name is the "Imperial Star Marines" a transliteration of what they're called in Vilani. However, the term is considered archaic in the Imperium; as a service they're known as the Imperial Marines in Anglic, all of their brochures/documents/communications refer to the organization as the Imperial Marines. Many of the oldest Marine units found around the core areas of the Imperium, still proudly refer to themselves as Imperial Star Marines in their unit names and livery. As the Imperium expanded, the Imperial Marines expanded as well but these newer units are simply referred to as the "Imperial Marines" and not "Imperial Star Marines."
3. I'll preface my comments on this in that I don't think there should be an Imperial Army; at least not in the way that later very heavily American-based Imperial military particularly in the late Rebellion-era and the Lorenverse/GURPSverse seems to have this Army / Marines split.
I'd really think the majority of the Imperial Marines would be more akin to the British Expeditionary Force of early World War I - a small, well-trained, well-motivated surface army that is infantry heavy used to fight planetside battles with the Imperial Navy being the primary stick of the Imperium.
They'd be superb infantry but overall a little light on armored and vehicular assets which are large, difficult to transport, and often just sit in a transport unused. This is why Imperial Marines have such an emphasis on grav-belts, battle dress, and FGMPs; infantry are flexible and are always useful in the situations where you want Marines in a way that things like trucks or tanks are not (yes, a fully equipped grav-belted, battle-dressed, and FGMP carrying Marine is expensive, at least you can use that Marine in all kinds of situations; you don't need a grav tank for a boarding action). Again, I'm not a big fan of the Imperial Marines having these companies of grav armor and meson artillery and so on. 1) It sounds too USMC. 2) It doesn't feel very futuristic and too contemporary. 3) I like the idea that Imperial Marines have an organizational weakness/trade-off that is considered acceptable because the Imperium doesn't have unlimited budgets (again, these ideally equipped units with special and heavy units for every occasion on their TO&E feel too much like a militarist's Mary Sue force) - those who hold the purse strings for the Marines are the Imperial Navy and the Navy feels that the Marines don't need that many tanks because they'll always have naval fire support overhead.
Although few like to admit it, the Imperial Marines would actually have different quality of troops within the organization - some would be better-trained and better-equipped than other units. None are truly considered "cannon fodder" (it's expensive to transport troops in starships - using expensive starships to transport cheap troops is idiotic) but some units definitely have more grav armor or are better trained to take on specialized or difficult roles and so on. The best units would typically historically have existed in the Imperial core regions which have never had a history of rebellion; these "politically reliable" units recruit and train mostly from worlds in the core areas, with a percentage of troops mixed in from other regions - sufficient to give variety of the unit but the troops from the more culturally homogeneous core region would dominate the unit's "culture" and "character" to ensure the desired qualities of the units remains "core." All Marine units are mostly are deployed in various frontier areas of the Imperium; but these "core" units are shifted from one hotspot to another following the expansion of the Imperium and thus have never truly "returned" to their home region - instead bouncing from one naval depot or base to another, rotating out personnel as necessary but are essentially eternally "on campaign."
