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May one desecrate Yom Tov for the sake of hearing the Shofar blowing?
One may not desecrate Yom Tov for the sake of blowing or hearing Shofar. This applies even regarding performing Rabbinical prohibitions for the sake of hearing Shofar as even in such a case the Sages forbade desecrating Yom Tov.
Examples of attaining a Shofar through performing prohibitions: Based on the above if a Shofar is on top of a tree or across a river and there is no other Shofar available nevertheless one may not climb the tree or cross the river to retrieve the Shofar. Likewise if a Shofar became buried in stones one may not remove the stones in order to uncover the Shofar [unless one moves the stones with an irregularity]. One may not leave the city limits [Techum] in order to hear Shofar. It goes without saying that one may not carry the Shofar from outside the city limits [in order for it to be blown in the city].
Asking a gentile to desecrate Yom Tov to attain a Shofar: The above law which prohibits performing even a Rabbinical prohibition applies only to having a Jew perform the prohibited action. However a gentile may be asked to perform any Rabbinical prohibition for the sake of attaining a Shofar. Thus one may ask a gentile to climb a tree or bring it from outside the city limits [Techum] and cases of the like. However it is forbidden to tell the gentile to perform Biblical prohibitions on his behalf. However if the gentile went ahead on his own and performed a Biblical prohibition in order to attain a Shofar then it is permitted to use it.
A Shofar that was made by a gentile on Yom Tov: Thus if a gentile made a Shofar on Yom Tov it is permitted for a Jew to use it for the blowing. Nevertheless this only applies when a gentile made a Shofar using a horn that he owns. If however he made the Shofar using the horn of a Jew then it may not be used unless there is no other Shofar available.
A Shofar that was brought by a gentile from outside the Techum: If a gentile brought a Shofar from outside the city limits on behalf of a Jew it is permitted to be used for blowing Shofar even by the actual Jew that it was brought for on his behalf. This applies even if the gentile traveled a distance of twelve Mil until he reached the city. Nevertheless one must beware not to move the Shofar more than four Amos from the area that he received it from the gentile [unless there is an Eiruv in the city]. [Likewise even if there is no Eiruv in the city if one received it in his house he may carry it anywhere within his house although he may not carry it outside.]
A Jew himself may never desecrate Yom Tov in order to attain a Shofar, even if it only involves a Rabbinical prohibition. However one may ask a non-Jew to perform a Rabbinical prohibition for the sake of attaining a Shofar. If a non-Jew went ahead on his own and did a Biblical prohibition in order to attain a Shofar, one may nevertheless use the Shofar.
May a woman ask a gentile to perform a Rabbinical command for the sake of her hearing Shofar?
No as a woman is not obligated to hear Shofar.
May one walk past the Techum to blow Shofar?
No. All the laws of Techum apply equally on Yom Tov as on Shabbos.
In an old age home or hospital, may one in need use an elevator to come to Shul to hear the Shofar?
Asking a gentile to press the button: If there is no other way to hear the Shofar some Poskim rule that a man may ask a gentile to press the buttons of an elevator for him/her to go down or up. It however requires further analysis if one may ask the gentile to press the button for him to go back up. A woman may not ask a gentile to press the button for the sake of hearing Shofar as explained above.
 The reason: The reason for this is because observing Yom Tov is both a positive and negative command as is written in chapter 498/34, and thus the positive command of blowing Shofar does not override the positive and negative command of Yom Tov. [ibid]
 Now although Shofar is a Biblical command nevertheless it is overridden by the Rabbinical decree. The reason for this is because the Sages gave their commands the same power as Biblical commands and hence they established that even their commands have ability to override Biblical commands in a situation that one is not required to actively nullify the command but simply to avoid performing it [Sheiv Veal Taaseh]. [ibid] In other words in cases that there is a Biblical command to perform an action and that action involves a Rabbinical prohibition, the Sages established that one is to refrain from fulfilling the action. However the Sages did not establish that one is to do a Biblically forbidden action for the sake of performing a Rabbinical command. [See also 128/3; 498 K”A 5; 628/5; 640/10]
 See 308/15; “The Laws of Shabbos” Muktzah Halacha 3A!
 586/24; See also 594/2
 Now although the gentile made the Shofar for the sake of the Jew [and we should hence prohibit its use due to that one may come to ask a gentile to initially make one for him] in this case we do not suspect the Jew will request from the gentile to make him a Shofar as this suspicion only applies by a matter which the Jew receives bodily pleasure from. However Mitzvos which were not given for one to receive bodily pleasure we do not suspect that one will transgress a prohibition in order to fulfill it. [586/24]
 As in such a case the prohibition of Nolad on Yom Tov does not apply being that the gentile owned the horn during Bein Hashmashos and the gentile did not remove his mind from making the horn into a Shofar [at that time] as the horn is within his possession and discretion to do with it as he pleases. [586/25]
 The reason: The reason for this is because the Shofar is considered Nolad on Yom Tov and hence is forbidden to be moved by a Jew even in order to use it for blowing, as we do not transgress [even] a Rabbinical command [of Muktzah] in order to fulfill the Mitzvah of Shofar [as explained above]. The reason why the Shofar is considered Nolad is because during the time of Bein Hashmashos the horn was not designated to be turned into a Shofar being it was not within the possession of the gentile. [586/25; See 495/13 that we are stringent to forbid Nolad on Yom Tov; See 252/12 and 325/6 that Nolad does not apply when the item is owned by the gentile. Vetzaruch Iyun as the second opinion there rules that it only does not apply if the Gentile began doing the work before Yom Tov and here it seems Admur plainly rules like the first opinion there. See also 308/23]
 As if there is no other Shofar available other than the one made by the gentile, one may be lenient like the opinion which rules that Nolad is permitted on Yom Tov in order so one not come to abstain from fulfilling the positive command in the Torah of blowing Shofar. [586/25; See 495/13]
 This applies even according to those opinions that rule that traveling with an object for more than 12 Mil from outside the city is a Biblical prohibition. The reason for this allowance is because an item that was brought from outside the Techum is permitted to be moved and is only forbidden in eating or benefit. Now although benefit includes the use of the object as explained in 515/1, nevertheless it remains permitted to blow the Shofar as one who blows the Shofar for the sake of the Mitzvah is not considered to be receiving any benefit at all, as explained in 586/24 regarding using a Shofar that was made on R”H. [ibid] However see Taz 586/4 that some benefit is received. Vetzaruch Iyun.
 As is explained in 515/15
 515/15; The rule is that although it is permitted to carry on Yom Tov without an Eiruv, an object that came from outside the Techum receives the same laws as Shabbos and hence may only be carried in areas that it is permitted to carry on Shabbos itself. [ibid]
 Shearim Hametzuyanim Behalacha 135/4; Nishmas Avraham 585 [p. 700];
 Shearim Hametzuyanim Behalacha 135/4; Nishmas Avraham 585 [p. 700]; Piskeiy Teshuvos 586/11; See Chelkas Yaakov 3/137
 The reason: As majority of Poskim rule that lighting a fire on Shabbos [i.e. electricity] is only Rabbinical in which case one may ask a gentile for the sake of a Mitzvah as explained above. [ibid]