(In order of your original question)
First of all, I would like to correct an apparent misconception regarding the halakhic positions of traditional Temanim. While it is true that they they generally follow the Rambam as opposed to later authorities, at no time in their history did the Jews of Yemen follow the Mishneh Torah exclusively. Instead, the previous Mara De-Athra of Yemen - i.e. before the time of the Rambam - was the Rasag (Rav Sa`adia Gaon) and even then they followed other Gaonic opinions in many areas of halakhah. While there were strains of "Rambamists" in Teman who attempted to exclusively follow the Rambam in all matters, the sentiment never truly caught on, even among the Dar Daim of early 20th Century.
(NOTE: When we speak of "traditional Temanim" in this regard, we are really talking about the Baladhim and not the later Shami'im who abandoned many of their ancestral traditions in favor of adopting Sefaradi practices based largely on the Qabbalah and the Shulhan Arukh of Rav Yosef Karo.)
Regarding the problems of modernity, the problem for the Mishneh Torah is not so different from the Shulhan Arukh. In actual fact, Rav Karo merely quotes the language of the Rambam in about 85% of his code. So, what is difficult for one is difficult for the other. However, from my experience, Temanim who still follow the Rambam attempt to derive - using logic and the Gemara - the shitah of the Rambam for more modern halakhic issues. Rav Ratzon Arusi shlit"a will often learn through the traditional sources, such as the Tur and the Shulhan Arukh, as well as modern poseqim like Rabbi Mosheh Feinshtein z"l in order to determine the correct halakhic position in his estimation.
In many instances, things which are "issues" for later codifiers of halakhah are simply non-issues in the halakhic formulation of the Rambam. So, they will simply disregard it and teach their communities that for the Rambam it was not an issue and that one is not required to worry about it.
While there are groups of traditional Temanim who do use electricity on Yomim Tovim, there are none who use electricity on Shabboth. Those who use electricity of Shabboth are not Temanim at all, but only "Rambamists" who generally take a simplistic and almost "Karaite-style" approach to the text of the Mishneh Torah. An approach which has led to a multitude or errors - some of which are rather severe - among non-Temani [would-be] devotees of the Rambam.
They handle the question in similar manner to other authorities. However, the premise(s) for determining any one position regarding date lines are taken from the Rambam as opposed to other authorities.
The answer to this question is essentially the same as above.
As for modern halakhic commentaries and responsa, you can find thousands of such questions and answers on the following websites (which are in Hebrew ONLY):
B. www.moreshet.co.il - there are many rabbanim who contribute to this site, so click on the "Ask The Rav" tab, and then select "Rav Ratzon Arusi" to see his pisqei teshuvah according to the Rambam.