According to Islamic Shari’ah, it is not permissible for a girl and a boy to have any kind of relationship with each other before marriage regardless of whether the relationship is direct or indirect (e.g. speaking), if it is with the intention of seeking pleasure or when it entails corruption and it is feared that they might fall into a sin. However, there would be no problem in conventional speaking with a non-Mahram provided that the two sides do not seek sexual pleasure and that they abide by the limits and the girl observes the Islamic dress code. Hence, you can speak with your cousin and have family relationship with him, if you observe modest dress and do not have lustful intention.
The answer to your question is as under:
1. In Islam, there is no such thing as "brother" and "sister" relationships outside of actual sibling relationships. In fact, brotherhood contract has been recommended among men, a step taken by the Messenger of Allah (S) following his migration to Medina. Obviously, the bond which was built on the basis of some commonalities among them was upheld and continued to be preserved. For example, the Holy Prophet (S) created brotherhood relationship between Abu Bakr and Umar; Uthman and Abdur Rahman bin ‘Awf; Talha and Zubair. The Prophet (S) himself established brotherhood relationship with Imam Ali (A.S.).[1]
When it comes to your contract with your cousin, it had been wrong in the first place.
2. You cannot shake hands with each other in the name of brother and sister relationships. There is no such a relationship to make you mahram to each other and it is, therefore, haram and sinful to shake hands with your cousin with the assumption that he is your brother .[2]
3. According to Islamic Shari’ah, it is not permissible for a girl and a boy to have any kind of relationship with each other before marriage regardless of whether the relationship is direct or indirect (e.g. speaking), if it is with the intention of seeking pleasure or when it entails corruption or when it is feared that they might fall into a sin.[3] However, there is no problem in conventional speaking with a non-Mahram without lustful intention provided that the two sides abide by the limits and the girl observes the Islamic dress code.[4] Hence, you can speak with your cousin and have family relationship with him, if you observe modest dress and do not have lustful intention.
4. Boy-girl friendship is not allowed because it is feared that they might fall into a sin. However, there is no problem in a job-related relationship, if it does not entail any corruption and that the Shari’ah rules are observed.[5] The conclusion is that you cannot make friendship with your cousin.
In a nutshell, you can speak with your cousin to the necessary extent and benefit from his guidance in ironing out your problems provided that you should observe the limits and abide by Shari’ah standards.
For further information, refer to answers 6912 (site: 6998), 6943 (site: 6933) and 1408 (site: 1428).
[1] - Extracted from answer No.6843 (site: 6933).
[2] - Extracted from answer No.2036 (site: 2092).
[3] - Extracted from answer No.1408 (site: 1428).
[4] - Gulpaigani, Sayyid Muhammad Reza Musavi, Majma’ul Masail, vol.2, p. 229, Dar al-Quran al-Karim, Qom, 2nd edition, 1409 A.H.; Mahmoodi, Sayyid Mohsen, Masail Jadid from the Viewpoint of Grand Religious Authorities, vol.1, pg.137-138, Sahib al-Zaman Cultural and Scientific Publications, Waramin, 14th edition, 1381 (en4766).
[5] - Extracted from answer No.1408 (site: 1428).