All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The right of the neighbor is great in Islam and being kind to the neighbor is among the perfection of faith. Abu Hurayrah narrated that the Prophet said: “Whoever believes in Allaah and the Last Day, he should be kind to his neighbor.” [Al-Bukhari and Muslim] On the contrary, the Islamic Law stated that harming the neighbor is among the signs of the weakness of faith; Abu Shurayh narrated that the Prophet said: “I swear by Allaah that he is not a (true) believer, I swear by Allaah that he is not a (true) believer, I swear by Allaah that he is not a (true) believer." The companions asked: "Who is it, O Prophet of Allaah? He said: "The one whose neighbor is not safe from his evils and mischief." [Al-Bukhari]
For more benefit, please refer to Fatwa 85376.
You should know that your situation with the neighbors who are good with you and who are bad with you is probably like everybody’s situation. Therefore, we advise you to be patient and wise when dealing with your neighbors who harm you. For instance, it is wise to use any permissible means in order to get rid of their harm, like being kind with their wives and children so that there would be a kind of friendship between you and them and a way to get rid of their harm. Muhammad ibn ‘Ali ibn Abi Taalib, who is known by the name “Ibn Al-Hanafiyyah” said: “He is not wise who does not deal kindly with those whom he has no other alternative except to deal with until Allaah makes a way out for him.”
However, if a person is not able to bear the harm of his neighbor, and the latter did not act according to his advice after being patient with him, then in this case, it becomes permissible for him to take the matter to the authorities if he does not fear harm [by doing so]. Abu Hurayrah narrated: “A man came to the Prophet complaining about his neighbor, and the Prophet told him: “Go and be patient [with him].” Then the man came a second time and the Prophet again told him: “Go and be patient [with him].” When the man complained the third or fourth time, the Prophet told him: “Go and take your belongings out into the street.” Therefore, the man did so. Then the people would ask him about what was going on and he would tell them his story. So, the people started to curse that neighbor: May Allaah …..May Allaah… (and they made Du’aa’ against him). Then his neighbor came to him saying: “Go back (to your home), you will never see from me that which you dislike again.” [Ibn Hibbaan]
Besides, if you can move to another house where you may find neighbors who are good to their neighbors, then do so.
As regards informing your neighbors that you did not find the shoe in your yard while actually you did find it, then this is a lie for which you should seek repentance from Allaah. Also, you are wrong by putting the shoe in the rubbish bin, because you are obliged to give it back to them even without their knowledge if the shoe is still available, and if it is not available, you should guarantee its price and pay it to them even indirectly.
Allaah Knows best.