Is the sacrifice made by the family of a resident in Yemen, whose expenses are paid by him as his agent, considered as such, and does it entail the obligations of the one making the sacrifice, such as refraining from cutting hair and trimming nails?

If you instruct your family in Yemen to purchase a sacrificial animal on your behalf, then you should refrain from taking anything from your hair or nails. If you have not intended the sacrifice or instructed it, then you are not considered a sacrificer and are not obliged to refrain from your hair and nails, because that is only required from one who intends the sacrifice.