The Bible says a great deal about testing prophets and spirits.
A spirit will confess that Jesus Christ has come in the flesh.
Dear friends, do not believe every spirit, but test the spirits to see whether they are from God, because many false prophets have gone out into the world. 2 This is how you can recognize the Spirit of God: Every spirit that acknowledges that Jesus Christ has come in the flesh is from God, 3 but every spirit that does not acknowledge Jesus is not from God. This is the spirit of the antichrist, which you have heard is coming and even now is already in the world.
1 John 4
Although this verse says 'spirit' I think it is just as easily applied to prophets. Both are charged with speaking God's words.
This verse also not only says it is okay to test the spirits but implies that you should every time.
A person speaking by the spirit will claim Jesus is Lord
3 Therefore I want you to know that no one who is speaking by the Spirit of God says, “Jesus be cursed,” and no one can say, “Jesus is Lord,” except by the Holy Spirit. 1 Cor. 12:3
A prophet will not contradict God's word
20 Consult God’s instruction and the testimony of warning. If anyone does not speak according to this word, they have no light of dawn. Isaiah 8:20
A prophet will not prove himself with wonders then ask you to worship other Gods
If a prophet, or one who foretells by dreams, appears among you and announces to you a sign or wonder, 2 and if the sign or wonder spoken of takes place, and the prophet says, “Let us follow other gods” (gods you have not known) “and let us worship them,” 3 you must not listen to the words of that prophet or dreamer. Deut 13:1-3
The predictions of a prophet must come to pass
20 But a prophet who presumes to speak in my name anything I have not commanded, or a prophet who speaks in the name of other gods, is to be put to death.”
21 You may say to yourselves, “How can we know when a message has not been spoken by the Lord?” 22 If what a prophet proclaims in the name of the Lord does not take place or come true, that is a message the Lord has not spoken. That prophet has spoken presumptuously, so do not be alarmed. Deut 18:20-22
These verses imply that an already proven prophet can later be unproven. This backs up 1 John's claim to test them every time and not just the first time. Apparently saying God said something that He did not is a capital offense, so you could argue that a prophet of ancient times was very slow to speak.
A true prophet will live a righteous life
15 “Watch out for false prophets. They come to you in sheep’s clothing, but inwardly they are ferocious wolves. 16 By their fruit you will recognize them. Do people pick grapes from thornbushes, or figs from thistles? 17 Likewise, every good tree bears good fruit, but a bad tree bears bad fruit. 18 A good tree cannot bear bad fruit, and a bad tree cannot bear good fruit. 19 Every tree that does not bear good fruit is cut down and thrown into the fire. 20 Thus, by their fruit you will recognize them. Matt 7:15-20
False prophets will predict Jesus' Second Coming
23 At that time if anyone says to you, ‘Look, here is the Messiah!’ or, ‘There he is!’ do not believe it. 24 For false messiahs and false prophets will appear and perform great signs and wonders to deceive, if possible, even the elect. 25 See, I have told you ahead of time. Matt 24:23-25
I suppose verse 25 makes Jesus a prophet because it has come to pass.
So the Bible does not only support the idea that prophets and the like should be tested but even tells you how. Furthermore, death was the common punishment for giving false prophecies. It was a very serious crime, therefore, we could argue the reverse that claiming prophecy was a very serious claim that required serious scrutiny. Logically, if a prophet were from God he would almost demand this testing because having God's word heard is his charge from God and the only reason he is speaking. So here I want to define a prophet, at least in the way I am using the word: A prophet does not necessarily predict the future. In fact, more generally, he has a message from God.
As far as following which prophet when two of them disagree, meaning their 'messages from God' are mutually exclusive, you first need to ask if they are speaking from the Lord. If they both say yes then at least one of them is wrong and the tests above should be applied and if either of them is wrong he should never be heard again. If one or both says no then it is clear that his words have only his own authority and you can ignore them if you choose.
Saying a prophet's words are infallible does not seem supported by these verses. However, once proven to be from God, it is logical to say that those words are equal with scripture; the same God made both the scripture and the prophet's message. The distinction should be made that a prophet's words should only be tested as from God if the prophet claims that they are from God; it does not make sense to test as holy that which makes no such claim. This is supported well when Paul made the distinction of his advice and the Lord's commands in 1 Cor. 7. These verses demand that we consider Paul a prophet, by the definition I provided, because he claims to be speaking from God. He is not making a prophecy, per se, but he does claim that the following commands are God's words. Our recourse would be to compare these verses to what is already proven as scripture. Fortunately, for Paul they do match. On all his other writings, however, he rarely made such direct claims so we are left with a small discussion contained in the comments below.
Your questions under note 1 are easily addressed with the verses above, especially the methods for testing. In the Old Testament it is clear that a prophet did not always speak from the Lord; not only would that be a terrible burden for him, but the very idea seems somewhat unscriptural because the people would hang on every word of the prophet rather than worship God and pray for His intersession. This is dangerously close to idolatry, in my opinion, which is strictly forbidden in numerous Old Testament texts.
However, your anecdote illuminates an issue. Your story does not mention that this particular person claims to be speaking from God. Furthermore, there is a difference between healing and prophecy; they are not the same, but you are correct in assuming that they can both be from God or not, as supported in Deut. 13. I submit that a healing is always good even if the motives are wicked. Logically then, if a person only heals and never claims to speak from God then there is no worry that he might be followed as a false prophet. But your story does say that he claims that testing if his healings are real (which there is no biblical way to do so) would lead you to suffer the same illness that was healed. This is a spectacular claim and if he does say it is from God then someone brave or skeptical enough should test the validity of the healing (I suppose with medical tests or something). If that person does not soon suffer that very illness as the 'prophet' claims then he fails at least one of these tests: What he said did not come to pass. (In my opinion things already seem fishy so I would gladly volunteer to perform the medical tests). I cannot find a verse that demands we test validity of healings but we should consider 2 Thes. 2:9 which warns that there will be many false signs and wonders in the last days and Matt. 12:22 where Jesus does not curse the pharisees that say His healings are by Beelzebub instead He merely corrects them, therefore, showing that it is at least odd to say you will suffer the same illness if you test the healing.