No. The Mishna Berura (12:2) writes that when the Tzitzit tear but remain valid for use, they are valid only Be'di'avad after the fact but one may not initially tie such Tzitzit on a garment.
The Mishna Berura (12:7) writes that if one tied the torn piece to the remaining part of the string before the Tzitzit were tied onto the garment, then the Tzitzit are valid even if less than four centimeters of the string remained when it snapped. If the strings snapped only after the Tzitzit were affixed to the garment, then one may tie the torn piece back onto the string only if the Tzitzit were not disqualified by the string's snapping. If the string snapped in such a manner that rendered the Tzitzit invalid, one must first untie the strings from the garment and only then tie the torn piece back onto the remaining part of the string. The strings are then considered valid and may be tied back onto the garment.
The first view cited by the Shulchan Aruch (12:1) would allow using the Tzitzit, since there is no one entire string that does not contain the minimum required length of "Kedei Aniva," which is assumed to be approximately four centimeters. The second view, however, would disqualify these Tzitzit, since it demands that at least two complete strings remain fully intact. The Shulchan Aruch (end of 12:1) rules that one should endeavor to follow the second, stringent view, though strictly speaking Halacha follows the lenient position. The Mishna Berura (12:13) explains that if one can easily find a set of Tzitzit that satisfy the stringent view, he must use such Tzitzit, but he need not trouble himself and certainly not miss the public prayer service for this purpose. The Mishna Berura also writes (12:11) that one may recite a Beracha over Tzitzit that are valid only according to the first view, but not the second view. In our case, then, one should try to replace the Tzitzit, but if no other Tzitzit are readily available, he may use these Tzitzit and even recite the Beracha.
These Tzitzit are valid according to all opinions. According to the first view cited in the Shulchan Aruch (12:1), the Tzitzit are valid because no string has less than the minimum required length of "Kedei Aniva." The second position would likewise validate these Tzitzit, because even if the two torn ends are from two different strings, and not the two ends of the same string, two complete strings remain fully intact.
As noted by the Rama (end of 12:1), the second view cited by the Shulchan Aruch would disqualify these Tzitzit. If the three torn strings are all on the same side of the knot, then clearly three of the four whole strings are not intact, thus disqualifying the Tzitzit. And even if one of the torn strings is on the other side of the knot, we cannot determine whether this string is the other end of one of the other torn ends, or the end of a different string. We must therefore suspect that three of the four strings are ripped, thus disqualifying the Tzitzit. The first view in the Shulchan Aruch, however, would allow using these Tzitzit. As mentioned above, in answer #3, one should endeavor to replace these Tzitzit, but if no other Tzitzit are readily available, one may use these Tzitzit and even recite the Beracha.
Yes, according to all views, because the other end of that string, on the other side of the knot, is still intact, thus satisfying the minimum required length of "Kedei Aniva."
Yes. Since the practice today is to ensure to keep each set of four ends of the Tzitzit strings together throughout the process of tying the Tzitzit, we can be sure that in this case, the other ends of the torn strings are fully intact, thus satisfying the minimum required length of "Kedei Aniva" for all four strings. (See Shulchan Aruch, 12:1.)
In this instance, we must be concerned that the two torn strings are the ends of the same string, in which case there is a string that does not meet the prerequisite length of "Kedei Aniva." The Tallit is therefore invalid, according to all views. (See Shulchan Aruch, 12:1.)
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