If Yazid was "shocked and cried" over the killing of Imam Hussain (as) by him men, he should punish his men or even suspended them from their post(s). But that was not the case. He didn't show any significant reaction over this. Anyway, I was curious about knowing the arguments of those who pray for him. But I guess they are not present over this forum.
Brother, now as you have asked about mourning over shaheed, here is how we believe it. Although, I am afraid Mr. moderator may feel bad and close this topic. Al Muhaddith Shah Abdul Haqq Dehlavi in Madarij un Nabuwat "When Holy Prophet (s) reached Madina, he saw that cries could be heard from most of the houses of Ansaar (the helpers) but not from Hamza's house. Holy Prophet (s) said that wasn't there anyone to cry over Hamza? The helpers (Ansaar) asked their women to mourn over Hamza first and then they may go and cry over their own martyr. The women went to Hamza's house in the evening and kept crying till midnight. When Holy Prophet(s) woke up and asked about it, he was told the whole thing. Holy Prophet(s) blessed them by saying" May Allah be pleased with you and your children." So you can see that mourning over shaheed is something that Prophet (sawa) himself recommended. And ofcourse, it is due to the love of Imam (as). And please correct your information. We do mourn death of Prophet (sawa) but not on 12th of Rabi-ul-Awwal. According to our history books it is 28th of Safar. And Imam Ali's shahadat is mourned on 21st of Ramadhan. You may inquire over the web as well. I hope this will lower your confusion regarding your concern.