At some point in the fall of the western half of the Roman Empire and the transition of the eastern half into the Byzantine Empire, the last remnants of roman military organizations and government institutions also disappeared.

The Roman military and government organizations didn't disappear, they just changed. Many of the traditions of the old Roman Empire survived in some form or another until the Fall of Constantinople in AD 1204, or arguably, AD 1453/1461. Also, the Byzantine Empire was officially known as the Roman Empire, and therefore should not really be treated as a separate entity, historians only really do this as a matter of convenience. Anyways:

what is the last recorded imperial Roman legion?

This is a difficult question to answer because first you have to define what you mean by "legion". If you're referring to the last use of the term legio to describe a battle-group in the Roman army, it is rather unclear. Vegetius uses the term in De Re Militari, which was written in the late 4th or early 5th Century, so we can assume that it was still widely used then. The term is used sporadically in accounts from the 6th and 7th centuries, but it definitely seems that the widespread use dwindles significantly. What is certain, however, is that the term "legion" to describe the armies of Byzantium is almost non-existent following the reign of Heraclius and the Arab Invasions (mid-7th Century). This is when I would say that the name legion no longer was used.

However, the legion was not really just defined simply by its term. Roman military organization was always adaptive and changing, and because of this, it makes defining what constitutes a "legion" difficult. By the late 3rd Century, the Roman legions were already changing significantly from what they had been during the early Empire. They went from being relatively large, all-purpose, offensively-focused units to smaller, reactionary, and highly specialized battle-groups. The infantry adopted lighter and more economical weapons and equipment, and the role of cavalry rose in importance. A large influx of Germanic peoples into the legions changed their identity and fighting tactics. Increasing hostilities between the Romans and a host of new enemies ensured that tactics continued to evolve significantly. By the start of the 7th Century, the Roman armies became virtually unrecognizable from the early Imperial legions. The formation of the themata under Constans II (AD 641-668) sealed their fate. This is where I would argue that any semblence of the system of the old legions definitively disappears.

When was the last meeting of the Senate?

The Senate of Rome? or just an Imperial Senate of the Roman State? The Senate in Rome actually survived the Fall of the Western Roman Empire and appears to have persisted until at least the early 7th Century, when it was disbanded by the unpopular Byzantine "usurper" Phocas, who felt the Roman Senate was a threat to his authority in the increasingly independent and sometimes hostile Exarchate in Italy, which was far from the direct authority of Constantinople and in reality under the control of the often-defiant Pope.

The Imperial Senate of Constantinople lasted, intermittently, until the Sack of Constantinople in AD 1204. However, by that time, it had been relegated to merely a position of ceremony, and served a very minimal role in governing the Empire.

Oh, and just one more thing:

We know the name of the last roman emperor, Romulus Augustulus

Romulus Augustulus was the last Western Roman Emperor, but even that's up for debate since Julius Nepos held the title (and was openly supported by the Eastern Roman Emperor Zeno) until his death in AD 480. There is very little reason to consider Romulus, the little Augustus, the "last Roman Emperor", especially since the Imperial capital since Constantine's time had been Constantinople, and the Empire remained strong in the East.

With that said, we can try to clarify the question of the "last Roman Emperor". Note, the true definition of "last Roman Emperor" has never been well-defined, so you'll always find people arguing for figures other than the ones I have posted here.

The last Roman Emperor who supported the old pagan traditions of the Roman Empire was Julian the Apostate, who unsuccessfully attempted to reverse the Christianization of the Empire during his reign (AD 361-363). Despite being a noted philosopher, he was killed in battle against the Persians, and his attempts to restore paganism fizzled out shortly thereafter.

Technically, the last Roman Emperor to reign from Rome was Carinus (AD 282-285). After Carinus was defeated, Diocletian moved his capital to Mediolanum, and the to Sirmium, and later changed the Roman political system to that of the Tetrarchy, which had four capitals (originally, Nicomedia, Sirmium, Mediolanum, and Trier), and even these weren't always set in stone depending on whether the tetrarchs were on campaign or not. I guess you could also argue Maxentius, but I don't really see him as an actual Emperor, since he was nominally bound to the rules of the Tetrarchy.

But, again, if we go by technicality, the last Roman Emperor to rule over an Imperial capital would be Constantine XI Palaiologos, who famously died in a final, glorious charge of defiance against the Ottomans during the Fall of Constantinople in AD 1453.