Noun/Pronoun agreements: The programmers or the aide will

May be the first one should use 'their'...rest are correct.

The president and treasurer is the same person. They would have been separate individuals if we had used like:

The president and the treasurer...

The president and treasurer gave his report.

Good one for practice

To answer first one:
1)The programmers or the aide will move their/his office today.: The verb must agree in number with the nearer noun. In this case aide( singular) is closer hence use singular verb( his).
2)Steve, in addition to his group, will give his/their reports tomorrow. :is a case of additive phrase: in addition to his group which does not change the number of subject. In this case Subject is Steve( singular) hence Use singular verb(His).
3)Neither of the women would admit her/their mistakes: The pronoun that comes after neither of is plural.Hence their.
4)he president and treasurer gave his/their reports. : Two singular subjects joined by and make up a compound plural subject hence plural verb their.
5)The faculty have been given its/their places on the roster.: plural verb have been hence the pronoun must be their.