If the corner of a tallis rips as a square (i.e with four edges of it's own, and more than 3 hand-widths, namely chayiv in tzitzis in it's own right) can it be reattached to the original garment with the (in-tact) tzitzis still on it?
Firstly, 3 hand-widths does not make a garment obligated in tzitzis; the size is 1x1 amot (Siddur Harav). If the cut-off piece is large enough to obligated in tzitzis in it's own right, it may be re-attached with the tzitzis still attached, and it's not a problem of תעשה ולא מן העשוי ("you shall make, and not use that which was already made") (Sulchan Aruch OC 15:3).
If it is not large enough to obligated in tzitzis in it's own right, it is the subject of a machlokes in Shulchan Aruch. Shulchan Aruch Harav (OC 15:8-11) gives a detailed account of the machlokes:
One opinion holds that the tzitzis as they are are pasul, since the piece they are attached to is not large enough to be obligated in tzitzis.
Another opinion holds that since it is attached to a piece of cloth that is large enough to be a corner (3x3 hand-widths), the tzitzis which are attached to it do not become pasul.
Shulchan Aruch Harav (se'if 11) paskens that lichatchila one should follow the first opinion, however, if that is impossible, e.g. one of the strings was torn (so it is not the length of 12 thumb-widths - the size of tzitzis string when putting them on initially) but still kosher, and other tzitzis strings cannot be found at all, one may rely on the second opinion.
See Gemara Menachot 41a:
אמר ר' יהודה טלית שנקרעה חוץ לשלש יתפור תוך שלש לא יתפור תניא נמי הכי טלית שנקרעה חוץ לשלש יתפור תוך שלש רבי מאיר אומר לא יתפור וחכ"א יתפור ושוין שלא יביא אפילו אמה על אמה ממקום אחר
A Talit corner which has been torn and is greater than 3 finger width, he may be reattached.
See in Shulchan Aruch (OC 15:4):
ורב עמרם פירש דטעמא משום דנקרע תוך שלשה לית ביה תורת בגד וכמאן דליתיה דמי ואף על גב דתפרה כמאן דפסק חשוב ואי עבד ביה ציצית לא פטרה לטלית
(This is not an exact translation but an explanatory paraphrasing)
Rav Amram that when the size of the detached piece is 3 X 3 finger width, this piece has cloth characters and the tsitsit which is attached to it remains a cloth tsitsit. Thus, when we reattach the piece we reattach an already called tditdot, not a non tditdit already made which becomes tsitsit by reattachment. All acharonimunderstand that Rav Amram eotds are true at least for a reattachment to the previous cloth.
Nimuke Yosef explained Rav Amram:
יתפור, שהרי לא נתבטל הציצית, כיון שנשאר שיעור שלה להיות כנף, דהיינו ג' על ג'.
The tsitsit nature of the detached tsitsit was not loosed.
In the Shulchan Aruch haRav, the Alter Rebbe rules:
"If each part of the Tallis retains a Shiur Tallis and contains four legal corners, one may sew the sides back together with their Tzitzis [or sew each side to another Tallis]. "
Shiur Tallis given as 6x6 cm, which I assume is the equivalent to the 3 etzbaot each way.