Is there a Biblical warrant to engage in debates with non-believers or cult members? What format did the apostle Paul used during his debates in the synagogues?

My personal opinion and practice is to avoid debates. Winning or losing tends to depend on the skillfulness of the debater, and not on the evidence. Paul speaks about avoiding the worldly "wisdom" and, in simplicity, preaching Christ (1 Corinthians 1:18-25). Paul depends upon the Holy Spirit to convict men of the truth (1 Corinthians 2). In 2 Corinthians 2:17 and 4:2 Paul speaks, once again, about avoiding worldly methods of communication. (Some, of course, would differ we me in my opinion that debating is a worldly/fleshly method.) In addition to the texts I have already mentioned, I would find the following texts to provide me with guidelines in this matter:

2 Timothy 2:14-26

14 Remind people of these things and solemnly charge them before God not to wrangle over words. This is of no benefit; it just brings ruin on those who listen. 15 Make every effort to present yourself before God as a proven worker who does not need to be ashamed, teaching the message of truth accurately. 16 But avoid profane chatter, because those occupied with it will stray further and further into ungodliness, 17 and their message will spread its infection like gangrene. Hymenaeus and Philetus are in this group. 18 They have strayed from the truth by saying that the resurrection has already occurred, and they are undermining some people's faith. 19 However, God's solid foundation remains standing, bearing this seal: "The Lord knows those who are his," and "Everyone who confesses the name of the Lord must turn away from evil." 20 Now in a great house there are not only gold and silver vessels, but also ones made of wood and of clay, and some are for honorable use, but others rather ignoble. 21 So if someone cleanses himself of such behavior, he will be a vessel for honorable use, set apart, useful for the Master, prepared for every good work. 22 But keep away from youthful passions, and pursue righteousness, faithfulness, love, and peace, in company with others who call on the Lord from a pure heart. 23 But reject foolish and ignorant controversies, because you know they breed fights. 24 And the Lord's slave must not be a fighter but kind toward all, an apt teacher, patient, 25 correcting opponents with gentleness. Perhaps God will grant them repentance and then knowledge of the truth 26 and they will come to their senses and escape the devil's trap where they are held captive to do his will.

James 3:13-18

13 Which of you is wise and understanding? By his good conduct he should show his works done in the gentleness that wisdom brings. 14 But if you have bitter jealousy and selfishness in your hearts, do not boast and tell lies against the truth. 15 Such wisdom does not come from above but is earthly, natural, demonic. 16 For where there is jealousy and selfishness, there is disorder and every evil practice. 17 But the wisdom from above is first pure, then peaceable, gentle, accommodating, full of mercy and good fruit, impartial, and not hypocritical. 18 And the fruit that consists of righteousness is planted in peace among those who make peace.

As I read the Book of Acts, Paul "reasoned" in the synagogues (see Acts 17:2-3; 18:19). I don't think this is debating, but a reasoned presentation, based upon the Scriptures.

My guess is that if Paul were asked the question you have raised he would say, "Preach the word. . . (2 Timothy 4:2). People are converted by the preaching of the Word, and they are also discipled in the same way.