Is it a valid observation that Iblees was only begun to be referred to as Shaytaan after he refused to prostrate to Aadam as ordered by Allaah? Also, how could Iblees have gotten access to the Garden again to deceive Aadam and his wife after he (Iblees) had been previously expelled by Allaah?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

The interpreters of the Quran differed in opinion with regard to how Iblees [the devil] entered Paradise, after Allaah told him (what means): {"Then get out of it, for indeed, you are expelled.}[Quran 15:34]. The most preponderant opinion of the scholars is that he was prevented to enter Paradise in a honourable and respectful way as the angels entered it, but he was not prevented to enter it being humiliated and debased, so that he may be used as a means of testing Aadam and Eve, may Allaah exalt their mention. For more benefit, please refer to Fatwa 90394.

As regards the name 'Satan', it is not specific to Iblees, but all the disbelievers among the Jinn are called Satans. An-Nawawi said: 'He was called Satan because of his pride and arrogance, and every disobedient, rebellious and arrogant person is a Satan, and it appears that this word "Shaytaan" is derived from Shatana, which means distant (in the Arabic language), if he is far from goodness and mercy.'

The name Shaytaan is also called for every arrogant human being, as Allaah Says (what means): {And thus We have made for every Prophet an enemy, devils from mankind and jinn.}[Quran 6:112] Ibn Taymiyyah said: 'Satans, are the arrogant human beings and Jinn.'

Based on the above, Iblees was named Satan because he refused to prostrate to Aadam, as he acquired those qualities. 

Allaah Knows best.