Since the early days of Christianity, wearing chapel veils has been a common practice among faithful women. Chapel veils, also commonly called mantillas, which comes from the word manta, meaning cape, are typically circular or triangular shaped pieces of black or white lace that are draped over a woman’s head when attending Mass, or in the presence of the Blessed Sacrament. Traditionally, the black veils were worn by married or widowed women, while the white veils were worn by young girls, or unmarried women. 
 
Throughout the centuries, the use of the mantilla by women has had many purposes. The wearing of the mantilla is an act of veiling a woman's physical beauty (as a woman’s hair is traditionally considered her crowning beauty), so that the beauty of God may be glorified instead. It is also a way of emulating Mary, our mother, who is the archetype of purity and humility. Moreover, the mantilla, or chapel veil, signifies the role of women as a life-bearing vessel. Oftentimes, the chalice holding the blood of Christ is veiled until the Preparation of the Gifts, and the tabernacle is veiled between Masses. Both of these vessels hold the Eucharist – the very life of Christ. In a similar fashion, the woman was endowed by God with the special gift of bearing new human life. Because of this, women, as all things holy and sacred, are veiled. 
 
Before the Second Vatican Council, the wearing of chapel veils was required for a woman when attending Mass, as a symbol of her modesty and humility before God. Although this practice is no longer required, it is still very much supported and encouraged by the Church as a sign of reverence and piety while in the presence of God. 
 
In addition to being a long-standing and beautiful piece of our Catholic tradition, there are three great reasons to wear a mantilla: (1) veiling is a physical reminder that you are entering a Sacred space, (2) veiling is a sign of feminine dignity, and (3) veiling is half of a dual-tradition whereby a woman covers her head (by covering her hair with a veil), and a man uncovers his head (by removing his hat) when entering a church as a sign of humility and devotion.

No, a man in the situation you describe could not continue to function as a permanent deacon. However, some distinctions are necessary so as to clarify the answer. 
 
While celibacy is not required of a married man who becomes a permanent deacon, celibacy does apply to some permanent deacons. And this can happen in a couple of different ways. First of all, if an unmarried man becomes a permanent deacon, he is required to promise celibacy at his ordination, and to remain celibate for the rest of his life. Secondly, if the spouse of a permanent deacon dies, he is expected to live celibately from that point forward. He is not to date, or seek a new spouse. 
 
In the unfortunate situation you describe of a deacon who is divorced, he also would be expected to live celibately from that point forward. This would be true even if his marriage received a declaration of nullity from the Church. 
 
In the thankfully rare situation where permanent deacons become divorced, the local bishop usually permits such a deacon to continue ministering as a deacon. But the bishop also needs to ensure that the deacon did all he could to reasonably save his marriage, and did not casually cast aside his marital vows, which would be a scandal. Presuming this can be assured and that scandal can be avoided, the bishop can permit a divorced permanent deacon to continue ministering. But, as already stated, he must live celibately from that point forward. 
 
What if the Deacon were to refuse to follow Church law, either by flagrantly divorcing and remarrying, or by remarrying after the death of a spouse? In such cases, he would be suspended from practicing his ministry as a deacon, and likely be laicized. Since ordination confers a character, he would still "be" a deacon, but could not, in any way, perform the ministry of the diaconate.