How can a Christian know if they are called to be a prophet, according to denominations that believe in modern-day prophets?

On the third question, many people misunderstand the "gift of prophecy." You should think in terms of "proclamation" (preaching) rather than predicting the future. The word "prophecy" throws people off. The gift of prophecy is not the same as someone called to the office of a prophet, or even only used for predicting the future.

Unlike the gift of tongues, which is in an unknown language and requires someone with the gift of interpretation of tongues to understand, the gift of prophecy is where the Holy Spirit gives an utterance in a language known to the hearer and thus does not require an interpreter.

People can experience the gift of prophesy without it having anything to do with predicting the future. As Paul explains the gift is for edification.

1Co 14:3-6

³ But he that prophesieth speaketh unto men to edification, and exhortation, and comfort. ⁴ He that speaketh in an unknown tongue edifieth himself; but he that prophesieth edifieth the church. ⁵ I would that ye all spake with tongues, but rather that ye prophesied: for greater is he that prophesieth than he that speaketh with tongues, except he interpret, that the church may receive edifying. ⁶ Now, brethren, if I come unto you speaking with tongues, what shall I profit you, except I shall speak to you either by revelation, or by knowledge, or by prophesying, or by doctrine?

On the question of how do you know someone is called to the office of a prophet, that's a good question. Pentecostal denominations do recognize God calls some to that office today. As far as predicting the future, scripture gives one evidence if the person is from God.

When a prophet speaketh in the name of the LORD, if the thing follow not, nor come to pass, that is the thing which the LORD hath not spoken, but the prophet hath spoken it presumptuously: thou shalt not be afraid of him.

Deut. 18:22

But the office of a prophet is a lot more than proclaiming the future. Prophets tend to be moral compasses whom God uses to correct the wrongs of society, especially the leadership. Prophets like Jeremiah and John the Baptist suffered greatly for making pronouncements against the rulers of their day. Jesus mentions the prophet Zachariah who was slain.

I will give a Lutheran continuationist non Pentecostal perspective on this question. The word for προφητεῖαι does not necessarily have a "gee whiz" or "thus saith the Lord" connotation. It can be a general word for speaking forth what God is revealing. In other words, it can be preaching or sharing in an applied manner what one believes God is bringing to mind for others to hear about from Scripture.

In the Lutheran tradition, the prophetic office continues in the form of the office of preaching (lay or clergy). A priest (i.e. pastor) prays and discerns God's guidance on what to preach about each Sunday. This practice can be traced to its devotional roots in the 16th century and long before that in catholic Lectio Divina (Latin for Divine Reading) traditions. For example, C. F. W. Walther in an article for “The Lutheran” back in 1847 wrote about a pastor friend of Martin Luther’s who observed him trying to hear from God on what to preach about:

...in the early years he had encountered Luther’s being so much in prayer the hour before he preached that he forgot about the preaching itself, and that on one occasion he had climbed into the pulpit with the words, “Don’t be surprised dear friends, I was delayed by God, with whom I have just had long discussion about the church, the university the city (of Wittenberg) and all of Christianity.

In Lutheran theology, prophetic hearing from God happens in, with and under the authority of the objective Word of Scripture. That is why Luther in his little book, "A Simple Way to Pray," speaks of prophetic gifts in the context of a discussion on the Third Commandment (i.e. don't take the Lord's name in vain).  In the book "Simple Way to Pray" Luther also writes about a two way process of communication that can often occur in the context of praying through the theological themes found in the Lord's Prayer. He writes:

If such an abundance of good thoughts comes to us we ought to disregard the other petitions, make room for such thoughts, listen in silence, and under no circumstances obstruct them. The Holy Spirit himself preaches here, and one word of his sermon is far better than a thousand of our prayers. Many times I have learned more from one prayer than I might have learned from much reading and speculation.

What Luther writes about is very parallel to what the Christian philosopher Dallas Willard writes out in his book, "Hearing God: Developing a Conversational Relationship with God." See especially pages 134-135.

The ultimate test is whether those claiming prophetic words in preaching help others by illuminating a Biblical Law/Gospel (faith, hope and love) Kingdom ethic of righteousness, peace (shalom) and joy - all under the authority of Scripture.