All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
A Muslim woman is not permitted to travel for work or for other purposes without being accompanied by a Mahram as the Prophet said: “It is not permissible for a woman who believes in Allaah and the Last Day to travel a distance of one night except if she is accompanied by her Mahram.” [Muslim]
Indeed, if there is a necessity, it is permissible for a woman to travel without a Mahram, but the woman in question may not have a necessity that renders what is forbidden to be permissible, as she is married and it is her husband who is obliged to provide for her needs. For more benefit, please refer to Fatwa 86565.
Nonetheless, travelling without a Mahram does not have any effect on the income if the work itself is permissible.
As regards working in customs and in collecting taxes, then this is subsidiary to the ruling on imposing customs and taxes; if they are imposed in an Islamically-legitimate way, it is permissible to work in it, otherwise no. For more benefit, please refer to Fataawa 85356 and 82663.
As regards Hajj and offering the ‘Eed sacrifice with unlawful money, then the person who does so is sinful but his Hajj and sacrifice are valid.
Imaam An-Nawawi said: “If a person performs Hajj with forbidden money or on a stolen animal, he is sinful and his Hajj is valid and sufficient according to the view of our school; this is the view of Abu Haneefah, Maalik and Al-‘Abdari, and it is also the view of most jurists. However, Ahmad said: 'It (his Hajj) is not sufficient (it does not free him from the obligation of Hajj).' Our evidence is that Hajj is specific acts while the prohibition (of those matters) is not related to.”
Moreover, Al-Qaraafi said in Anwaar Al-Burooq fi Anwaa’ Al-Furooq: “If someone prays in a stolen garment or performs ablution with stolen water or performs Hajj with unlawful money - all of these (acts of worship) are valid to us, contrary to Ahmad (ibn Hanbal). The basis for that is that the essence of those obligations, namely Hajj, covering the Awrah (during prayer) and purification, are achieved in the sense of fulfilling its benefit, but not through the legitimate means; and if the obligation is achieved in the sense of fulfilling its benefit, then the prohibition related to the means is a separate matter which involves trespassing against others, like the case of an unlawfully acquired house…”
Allaah Knows best.