Is it religiously obligatory for me to buy a plane ticket for my wife to visit her family, and would I be sinful if I did not? Am I obligated to buy her a phone when there is an agreed-upon monthly allowance? And does the payment of the monthly allowance continue during her travel period?

A husband is not obliged to buy his wife an airplane ticket to visit her family, and he does not sin by refusing, because her transportation and travel expenses are not his responsibility. Similarly, he is not obliged to buy her a new phone. The wife's right to maintenance (nafaqa) and equitable division of nights (qasm) is forfeited if she travels for her own needs with her husband's permission, though there is a difference of opinion among scholars regarding the continuation of maintenance if the husband permits it. It is preferable and better for the husband, if he can afford it, to buy her an airplane ticket, give her maintenance for the duration of her travel, and buy her a new phone. This is considered good companionship, noble character, and benevolence.