The answer to both questions is Yes.

We must qualify this answer to say that any such “appearance” is necessarily tailored to the cognitive capability of the recipient. God is infinite Spirit. The second Person of the Trinity took on a human nature in the Incarnation. This human nature is an essential dimension of God incarnate although not of the Godhead. If God the Father manifests himself to a human being, it will necessarily have to be through some kind of vehicle – whether it be a burning bush, a voice in the sky or in the form of a human being. Such vehicles are not “part” of the Godhead but are simply instruments used for the purpose of the revelation.

Let us start with the Old Testament. When the New Testament talks about God, the reference is to the Father. For instance, when St. Paul says, “The grace of the Lord Jesus Christ and the love of God and the fellowship of the holy Spirit be with all of you,” (2 Corinthians 13:13), he is referring to the Father when he says God. And God as Father in this and similar contexts in the New Testament is the same God who interacted with the people of Israel in the Old Testament. Thus, the direct encounters with God in the Old Testament are, for the most part, encounters with the Father (although there are also some that are clearly encounters with the Son and the Holy Spirit).

Here are some instances of “appearances” of God in the Old Testament:

“When Abram was ninety-nine years old, the LORD appeared to Abram and said: I am God the Almighty. Walk in my presence and be blameless.” (Genesis 17:1)

“The LORD appeared to Abraham by the oak of Mamre, as he sat in the entrance of his tent, while the day was growing hot. Looking up, he saw three men standing near him.” (Genesis 18:1-2)

“Jacob named the place Peniel, “because I have seen God face to face,” he said, “yet my life has been spared.”” (Genesis 34:31)

“When the LORD saw that he had turned aside to look, God called out to him from the bush: Moses! Moses! He answered, “Here I am.”” (Exodus 3:4)

“Since then no prophet has arisen in Israel like Moses, whom the LORD knew face to face.” (Deuteronomy 34:10)

The New Testament shows us direct encounters with the Father at the baptism of Jesus and at the Transfiguration:

“The holy Spirit descended upon him in bodily form like a dove. And a voice came from heaven, “You are my beloved Son; with you I am well pleased.”” (Luke 3:22)

“While he was still speaking, behold, a bright cloud cast a shadow over them, then from the cloud came a voice that said, “This is my beloved Son, with whom I am well pleased; listen to him.”” (Matthew 17:5)

“For he received honor and glory from God the Father when that unique declaration came to him from the majestic glory, “This is my Son, my beloved, with whom I am well pleased.” We ourselves heard this voice come from heaven while we were with him on the holy mountain.” (2 Peter 1:16-17)

These encounters did not end with the age of the Apostles.

In Christian history, the most famous direct appearance of the Father was witnessed by St. Ignatius of Loyola, the founder of the Society of Jesus, the Jesuits.

This is memorably described in the Protestant publication Christianity Today:

“During this time Ignatius had one of his most decisive visions. While in prayer one day, he saw Christ with a cross on his shoulder, and beside him was God the Father, who said, “I wish you to take this man [meaning Ignatius] for your servant.” Jesus said to Ignatius, “My will is that you should serve us.” Ignatius was also told that his group was to be called “the company of Jesus,” that they were to be like a company of fur traders yet focused on doing God’s will.”1

Here is another account:

St. Ignatius “took his way towards Rome, on foot, with F. Faber, and F. Laínez in his company [. . .] one day drawing nigh to the city of Rome, leaving the two Fathers in the field, he went into a deserted and solitary church some mile from the city to pray. There amidst the greatest fervor of his prayers, he felt his heart changed, and God the Father appeared to him, together with his most Blessed Son, who carried the Cross upon his shoulders and with the eyes of his soul, illustrated with that resplendent light, he saw that the Eternal Father, turning to his only begotten Son, commended Ignatius, and those in his company unto him, with exceeding great love, putting them into his hands.”2

St. Margaret Mary of the Sacred Heart and Sister Lucia of Fatima had visions of the Trinity and of the Father as a distinct Person. St. Margaret Mary describes the experience in her autobiography: “The Eternal Father presented me with a very heavy cross beset with thorns and surrounded with various instruments of the Passion and [the Father] said to me:  ‘See, My daughter, I make thee the same present which I made to My Beloved Son.'”  St. Faustina Kowalska of the Divine Mercy received direct messages from the Father.

So, to repeat, the answer to the questions with which we started is YES.

But how is an appearance of the Father possible? St. Thomas Aquinas shows us why there is no theological obstacle to its possibility.  But he points out whereas the Son and the Spirit are “sent” by the Father and manifest themselves as sent, the Father can manifest himself but not as one who is sent.

“The dilemma is that the missions follow the order of the processions (as we said), but the Father proceeds from none. The answer is that the Father is not sent, but is present in the soul by virtue of the Trinitarian concomitance we averred to in the last paragraph: “As to the Father, though He dwells in us by grace, still it does not belong to Him to be from another, and consequently He is not sent.”3

The visible mission adds something to the apparition, to wit, the authority of the sender. Therefore the Son and the Holy Ghost who are from another, are said not only to appear, but also to be sent visibly. But the Father, who is not from another, can appear indeed, but cannot be sent visibly. 4

End Notes

1https://www.christianitytoday.com/history/people/moversandshakers/ignatius-of-loyola.html

2https://www.uni-uenster.de/imperia/md/content/mittellatein/forschung/intersections/22alison_c_fleming_-_st_ignatius_of_loyola___s____vision_at_la_storta____and_the_foundation_of_the_society_of_jesus.pdf

3Summa Theologiae Ia Q. 43, A. 5.

4https://www.ccel.org/ccel/aquinas/summa.TP_Q39_A8.html