Question:May one buy and wear new clothes, do work on his house, and recite Shehecheyanu during the Sefira period? (I have recently been hearing that this is forbidden.)

Answer:The halachot of aveilut (mourning) for a deceased relative and the national mourning over the destruction of the Beit Hamikdash, during the days/weeks before Tisha B’av are discussed in the gemara. The minhagim of national mourning over the death of Rabbi Akiva’s students are not found in the gemara. There are both overlap and differences in the details for these different time periods.
Regarding the aveilut of the Sefira period, the Shulchan Aruch (Orach Chayim 493) cites prohibitions on marriages and hair cutting (and work after sunset – which is not widely accepted). The Mishna Berura (493:3) mentions the minhag of not dancing, which many have applied to all forms of instrumental music (see Igrot Moshe, OC I:166). These standard sources make no mention of the things about which you inquired.
Let us look briefly at minhagim about Shehecheyanu, clothes, and work on the house, as they appear in regard to the period before Tisha B’av. One is to reduce certain activities before Tisha B’av, including building projects (Yevamot 43a), but according to the Shulchan Aruch (OC 551:2), this is only during the Nine Days and not the Three Weeks. There is also a recommendation, which not all accept (see opinions in Mishna Berura 551:98), not to recite Shehecheyanu during the Three Weeks (Shulchan Aruch ibid. 17). The logic is that Shehecheyanu expresses our gratefulness to have made it to "this time," which may not be appropriate at a particularly sad time on the calendar.
While the standard sources do not mention these issues during the Sefira period, there are some sources that do, especially in regard to Shehecheyanu (see several opinions cited in Bein Pesach L’Shavuot 16:(2)). There is basis for this extension on two grounds. First, there is logic, as this is a nationwide sad period (as opposed to aveilut over a relative, which is personal- see Mishna Berura 551:98). Secondly, it is relatively easier to transfer minhagim when there is a model for such halachot, by doing, so to speak, "copy and paste" from one time to another.

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However, the logic and the model are also reasons, paradoxically, to ignore the minority strict opinions and the practice of some to refrain from some or all of the matters you mentioned, for the following reason. People can get confused as to what practices apply when. They remember that there is a concept of not saying Shehecheyanu and not doing renovations during national mourning periods, and they may have heard of someone knowledgeable who says to act this way during Sefira. They then may start adopting the practice, not based on a decision with knowledge of the sources and indications and a desire to accept the stringency. Rather, they think these are the standard minhagim. This is called a minhag ta’ut. In such a case, even one who has already followed the stringent practice may suspend it without hatarat nedarim.
Rav Ovadya Yosef has an interesting approach to these questions. First he explains (Yechaveh Da’at I:24) that one cannot call Sefira, which is actually the bridge between the joyous holidays of Pesach and Shavuot, a tragic period of time, as we call the period leading up to Tisha B’av. Therefore, he is against refraining from Shehecheyanu on fruit at that time. He is not against the stringency to avoid wearing new clothing that warrants Shehecheyanu, out of extra mourning. Regarding moving into a new home or doing work on an existing one, he simply permits the matter (ibid. III:30). The Tzitz Eliezer (XVIII:41) is perhaps more resolute in rejecting the appropriateness of stringency in these matters.
So, one need not be stringent and if he has been, he may continue if he likes, but he should consider whether his (family’s) practice is more based on confusion than a conscious decision to accept minority stringencies.