Is It Prohibited to Kiss Your Wife in Public?
Question

I am in alot of trouble. I can not ask local Iman because he is not good in his knowledge and I am embarrassed to ask friends or family. You see, the problem is I got married recently and I think I might have committed zina with my wife. This is how we were driving my sister's car and me and my wife stopped the car in a quiet place and we kissed a lot, quite a lot. a man walked passed and he saw us, but we did not stop. I feel I have committed open zina. Do we need to pay Kafarah (expiation), or confess to our elders? My second very very important question is after kissing, I let out (we were both wearing clothes all the time mufti sab) sperm in my underwear. Now I feel very bad because this is my sister's car and she has been driving it after me. i can not wash the car seat as she will get suspicious and confront me. i do not want to lie, now she sits in the car sits on the floor to eat dinner, sits on the sofa, her and her children and i feel very very bad as to what to do. is the car and sofa and place were we sit not clean anymore, and my wife and i are arguing she blames me for worrying. i think she is a bad muslim, because she does not fear Allah, because she made me stop the car and kiss, i hate her for that. Dear sir, I await your reply as I am disturbed by what has happened, i like to keep pak and clean at all times to please Allah.
Praise be to Allah.The nature of human imperfection
Please note that you will never be able to live your life without sins and mistakes. This is the nature of man. The Prophet (peace and blessings of Allah be upon him) said: “Every son of Adam commits sin, and the best of those who commit sin are those who repent.” (Narrated by At-Tirmidhi, 2499 and Ibn Majah, 4251. It was classed as sound by Al-Albani in Sahih At-Tirmidhi)
This Hadith clearly indicates that man will inevitably commit sin, but what matters after that is what he does about his sin. The believer repents to Allah from all his sins, gives them up and asks Allah for forgiveness every time he commits a sin, and he regrets doing it and resolves sincerely not to do it again. If you do that, then understand that Allah is Oft-Forgiving, Most Merciful; He forgives all sins for the righteous believer who repents sincerely and admits his sin humbly before Allah. Allah says (interpretation of the meaning): {Say: O My slaves! who have transgressed against themselves (by committing evil deeds and sins)! Despair not of the Mercy of Allah, verily, Allah forgives all sins. Truly, He is Oft-Forgiving, Most Merciful.} [Az-Zumar 39:53]
Is kissing one’s wife in public Zina?
The sin that you have admitted to is not Zina with your wife! Rather it is kissing her in front of other people, because there is no Zina with one's wife, rather Zina is done with a woman whom it is not permissible for a man to touch . But in the case of a wife, it is permissible to touch her. 
Is it prohibited to kiss your wife in public?
It is not permissible for a man or a woman to speak to anyone else of what happens between them in bed which no one else should see but them, because of the evils and temptations that result from that and because it opens the door to the devil. This has to do with one who speaks about what he did with his wife, so how about one who does things in front of people where they can see him! 
Shaykh Muhammad ibn Ibrahim (may Allah have mercy on him) said in his Fatawa (10/277) that kissing one’s wife in front of people is not permissible . 
Expiation for kissing one’s wife in public
As for the expiation for this sin , there is no expiation except sincere repentance and firm resolve not to do it again, and true regret for having committed this sin. 
As for confessing to your parents, there is no need for that. The sin that you have admitted to is a sin against Allah, so confession should be made to Allah. It is between you and Allah and you should not tell anyone about it, but be sincere in your repentance to Allah and Allah will forgive you. Allah is Oft-Forgiving, Most Merciful. 
The fact that your wife is the one who told you to do this deed does not mean that she is not a righteous woman or that she does not fear Allah. You also agreed with her to do it, and you did not stop even when that man saw you. So you should also shoulder your share of responsibility for what you did. 
May Allah help you to do that which He loves and which pleases Him. 
And Allah knows best.