What the Devil looked like
The answer is not clear from the biblical texts of Jesus' temptation in the wilderness, as no description of the devil's appearance is given. Other biblical texts give various descriptions which may or may not be intended to define his actual appearance:
- In Gen. 2 he is a serpent, but one that has legs.
- In Revelation 12:9, he is “the great dragon” and “ancient serpent.”
- In 1 Peter 5:8 he is compared to a "roaring lion."
- In 2 Corinthians 11:14 he can disguise himself as "an angel of light."
Other demons mentioned in the bible took various forms. The "se'irim" were hairy, satyr-like beings. (Isa. 13:21, 34:14; Lev. 16:10) The "shedim" (Deut. 32:17) were storm-demons, represented in ox-like form. Lilith was a female demon, a succubus, described sometimes as a desert owl (Is. 34:11) and possibly the mother of Tubal-Cain in Gen. 4:22. source
We can imagine that Satan, as the chief of the demons, could assume any of these or countless other forms. The question becomes which form would be most likely to succeed in tempting Jesus. For me the answer would not be one of the frightening forms but one of more attractive ones, and not necessarily the same form for all three temptations.
Thoughts or Real?
The OP also asks if the temptations were " strictly coming to Jesus through Satanic-driven thoughts." The first temptation (hungers) was physical. But the devil is described as tempting Jesus to perform a miracle and turn stones into bread. The other two miracles involve supernatural travel similar to what several other prophets experienced. The Gospels do not describe any of these as merely happening in Jesus' mind. Rather they involve "real" experiences of the spiritual realm. Believers may interpret them in other ways, but the text itself describes them as real occurrences, even though they may not have happened in the physical world.
Since the devil's purpose was to tempt Jesus rather than frighten, the most likely form he would take would be that an angel of light, a beautiful seductress, a wise teacher or other attractive form, rather than a dragon, satyr or serpent. Since two of the temptations involve supernatural travel (a high mountain and the pinnacle of the Temple) it is likely that these took place in the spiritual realm: not merely "satanic-driven thoughts" but substantial spiritual realities.