Shaking Hands with a Non-Mahram

Assalamualaikum Shaikh,I want to inquire you about the hand shaking with womens in non-muslim culture. Is it permissible in Islam to do Hand Shaking of men and women similar to Hand shaking between two mens as we normally do. 1. Some time practicing muslims are shaking hands with womens in front of public on the occasion of gatherin gs, meetings and conferences. I currently live in Canada which is predominently a non-muslim state here they have their own cultural customs and practices. As per their culture shaking hands, hugging and kissing of opposite sex is absolutely right specially at happy and sad occasions. 2. Can a man do hand shaking to women with the sole intention of dawah. As this kind of gesture and action can create a soft corner into a heart of non-muslim. If its allowed then does it incompliance to sunnah of Prophet SAS?. 3. Some brothers and sisters has this opinion here that we should not leave an impression of alianation into the hearts of local non muslims by not doing hand shaking or not wearing the cloths with their national symbols flags etc. Else they would hate us and we will not be able to talk to them about Islam. 4. Most of known dawah groups in middle east and asia when they come to North America they change their policies drastically as compared to back home, as per them just for the sake of Dawah.I know my questions are little bit tough but need an straight forward and Shariah based reply. No diplomatic answer please. Please give a sharia opinion with support of Quran and true Ahadith.Jazakallah KhairMohammed Fazil Ali