1. If the straps of one's tefillin are kosher but old, is it "hidur" mitzvah (advisable "beautification" of the mitzvah) to buy new ones?
Yes, one should buy new ones (R. Yosef Shalom Eliashiv, VaYishmah Moshe 36).
2. Is it "hidur" mitzvah to have tefillin straps that are black on both sides?
According to the Rambam, yes (Mishnah Torah, Laws of Tefillin 3:14). However, none of the other Rishonim rule this way, and the Knesses HaGedola says that regarding this matter, the custom is not like the Rambam.
According to R. Yosef Shalom Eliashiv, there is no need to have such straps, but it is permitted (VaYishmah Moshe 32).
On the other hand, there are two practical advantages to having them: the color is more long-lasting; also, one does not need to continuously check the position of the straps of the head tefillin as they hang down from the knot at the back of the head, to make sure that, at all times, it is the black side that is facing outwards.
3. What is the required length of the straps?
The strap of the arm tefillin must be long enough to wrap it around the arm seven times and afterwards tautly extend it the middle finger and wrap it around that finger three times.
The strap of the head tefillin must be long enough to encircle the head, tie the knot, and have the emerging part hang down to the navel (Shulchan Aruch, Orach Chaim 27:11).
For the head tefillin, the custom is to have longer straps, so they hang down farther, at least to the waist, and the custom is also that the right-hand strap be a bit longer than the left-hand strap.
4. What is the halacha of the part of the strap that is beyond the required minimum length?
The Mishnah Berurah (33:3, Biur Halacha) is uncertain whether it is halachically regarded just like the main part of the strap, regarding the black coloring, chatzitza and other matters.
5. What is the required width of the strap?
According to the Chazon Ish, the width must be eleven millimeters. According to R. Chaim Noy, it need be only ten millimeters. In a pinch, nine millimeters is sufficient (even according to the Chazon Ish). If it is any less, one should put on the tefillin without the blessing. Due to twisting during usage, if a strap's width is less than the minimum but if one pulls it taut, it will have the minimum width, it is kosher.
6. What is the halacha if a tefillin strap is not all black?
On the strength of the principle, "Most is like all," it remains kosher (Mahrshag 1:7).
Some have a doubt about this, so it is preferable to make sure that all of it is black (Daas Kedoshim 33:3).
7. Is a permitted to take straps that were used for Rabbeinu Tam tefillin and put them on Rashi tefillin?
It is forbidden, because of the dispute as to which type possesses the higher level of kedusha. (Shulchan Aruch, Orach Chaim 42:1).
For the same reason, it is forbidden to use the bag of one type for the other type, and so, too, the casings (Mishnah Berurah there).
If straps of both types get mixed together, one is allowed to take the strap of his choice (Keren L'Dovid 9).
8. Is one allowed to turn a strap around and re-tie it so that the end that beforehand was below is now above?
It is forbidden, because it lessens the kedusha of the end that beforehand was above (Kesset HaSofer 25:5).
9. What about a blue strap whose hue is so dark it looks black?
So long it appears to the eye as black, the halacha considers it black, too, and it is kosher (Biur Halacha 32:3). It is the same regarding other colors when their hue appears black (Chakel Yitzchak, Responsa 5).
10. Is one permitted to take part of a strap that once was used for wearing tefillin and use it in a different way for the mitzvah of tefillin? For example, can one cut a off a piece and wedge it into the base of one's tefillin so that the emerging strap remains firmly in place and does not wiggle? Or, can the piece be put into the box of tefillin of the arm so that the parchments remain in place and do not fall?
It is preferable to be stringent and not do so, but some authorities allow it, so he who is lenient has on whom to rely (Shevet HaKehati 3:52).