Touching a woman does not invalidate Wudhu in any circumstances, according to the most authorative opinion of the scholars. This is because it has been authentically reported from the Prophet salallaahu 'alaihi wasallam that he used to kiss some of his wives, and then he would pray, without performing Wudhu. [Abu Dawud nos. 178, 179 and At Tirmidhi no.86].
A woman should not shake hands with any man who is not a Mahram for her [mahram: her husband and any other man that she cannot marry, such as a brother, uncle etc.], just as a man should not shake hands with a woman for whom he is not mahram, based upon the saying of the Prophet salallaahu 'alaihi wasallam:
Åöäøöíú áÇó ÃõÕóÇÝöÍõ ÇáäøöÓóÇÁ ÔöÆúÊó
"Verily, I do not shake hands with women." [An-Nisaa’i no.4181 and Al-Muwatta no.2]
And what has been authentically reported from A’isha, may Allah be pleased with her, that the Prophet salallaahu 'alaihi wasallam used to accept the pledges of allegiance from women in words only; she said:
‘And no woman’s hand ever touched his hand.’ [Al-Bukhari nos.2713, 4891, 5288, 7214 and Muslim no.1866].
And Allah the most high says:
áóÞóÏú ßóÇäó áóßõãú Ýöí ÑóÓõæáö Çááøóåö ÃõÓúæóÉñ ÍóÓóäóÉñ
"Indeed, in the messenger of Allah (Muhammad salallaahu 'alaihi wasallam) you have a good example to follow..."[Al-Ahzab 33: 21]
Also, women shaking hands with men and men shaking hands with women whom they are not closely mahram for is a cause of fitnah for all. And the Islamic law which is perfect, prevents all things which lead to that which Allah has forbidden.
From the above, it is clear that the ajnabiyyah woman is one whom it is not forbidden for him to marry due to kinship or any other legal reason – this is the ajnabiyyah. As for the one whom he cannot marry due to ties of kinship, such as his mother, his sister, his aunt, or any other legal constraint, such as breastfeeding, or relationship by marriage, she is not an ajnabiyyah. And with Allah is success.