What form did Satan assume when tempting Christ? Or were these temptations strictly coming to Jesus through Satanic-driven thoughts?
Where is there any basis for claiming that Jesus Christ had any satanically-driven thoughts?
Yes, Satan tried to put thoughts into Jesus' mind that had all the hall-marks of the deceiver. Yet the Christ knew all about this evil one from before the time of the garden of Eden. He instantly recognised who was at work, 40 days into his wilderness fast, because although his body was weakened by that, spiritually he was untouched.
Matthew's account begins by giving us this insight -
"Then was Jesus led up of the Spirit into the wilderness to be tempted of the devil." Matthew 4:1 K.J.V.
I understand the Greek word translated 'led' has the forceful meaning of 'driven'. But as Jesus had seen the Holy Spirit descend in the form of a dove when "the heavens were opened unto him" at his baptism, and Jesus heard the voice of the Father declaring his pleasure at him, his Son, he went immediately into the wilderness fortified, knowing what was coming. He knew the purpose of the wilderness at that time. We may be sure that he used the 40 days to prepare himself spiritually for whenever the Devil made his move.
The Bible nowhere describes any form the Devil took, and nowhere tells us how Jesus saw him - whether literally, or spiritually. The simple answer to your question is "We do not know."
If we needed to know, that would have been included in the Holy Spirit inspired gospel accounts.
What we do know, from everything that is said about Christ, is that his thoughts were always driven by discerning the will of the Father (that he might do the will of the Father). That is stated time after time. His thoughts were never driven by the Devil.
Yet when Jesus went into the wilderness to fast for 40 days, the Devil must surely have found himself being tempted to try to test the Son of God at his weakest physical point, and he just couldn't resist it. The Devil's thoughts are always driven by devilish desires; Christ's thoughts are always driven by God's desires. And ne'er the twain shall meet.
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 (hunger) was physical. The devil is described as tempting Jesus to perform a miracle and turn stones into bread. The other two temptations involve supernatural travel, similar to what several other prophets experienced. Ezekiel was taken in the spirit to a valley (Ezekiel 37:1) Isaiah was spiritual transported to the Temple and God's throne. (Isaiah 6:1) The Gospels do not describe any of these temptations 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 did not happen in the physical world.
Since the devil's purpose was to tempt Jesus rather than frighten him, the most likely form he would take would be that an angel of light, a prince, a priest, 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: they were not merely "satanic-driven thoughts" but substantial spiritual realities.
Wilderness temptations of Christ: how did Satan appear to Jesus?
The Scriptures do not tell us what form Satan appeared to Jesus or what he looked like. Thus artists have a certain amount of freedom to portray Satan in several ways.
Supernatural knowledge in which the mind receives an extraordinary grasp of some revealed truth without the aid of sensible impressions. Thus St. Ignatius Loyola (1491-1556) wrote of his seeing "the humanity of Christ with the eyes of the soul."
These visions take place either through ideas that are already acquired and that are then co-ordinated and interpreted by God, or through infused ideas, representing divine things, that are thus better perceived than a person would otherwise perceive them. At times the visions are obscure and their object is only dimly understood; at other times the perception is very clear but lasts only a moment. The mystics describe them as intuitions that leave a deep impression on the mind.
The experience of St. Paul on the way to Damascus was at once sensible, imaginative, and intellectual. He beheld the blinding light with his eyes; he saw with his imagination the personal traits of Ananias; and his mind understood the will of God (Acts 9:3-12).