Q.1 The way 交通 is used in the past may be different from what we use nowadays, it is mainly used in describing traffic. However, if you read from old chinese literature, 交通 can be used in describing roads that are connected or linked such as “阡陌交通” from 陶渊明《桃花源记》. It may also be describing figurative speech such as "山川涸落，天气下，地气上，万物交通" from 《管子·度地》which implies everything around the world is linked to each other. These are only used in literature and are rarely used.
交流 means a transfer of information between two parties. It can be used in many different cases, such as 艺术交流, which means a interaction between people about art, 学术交流, which means the discussion of academics between people. It is mainly used when it is with a group of singular people discussing on a single topic or it may be two different groups of people interacting with each other. A word that could be used together with 交流 is 切磋, which in this case, means the clash of ideas.
联系 is usually used in somewhat formal settings. It might be people exchanging phone numbers for work then saying ‘我们以后会再联系联系’, meaning that they might call them later when they have spare time and he/she might want to sell a project to the opposing party. So it isn't necessarily a long period of time. However, in casual talk, it usually implies a long period of time where you have not been in contact with someone, such as “我们真的很久没有联系了”. So depending on where and how you use it the time period may be long or short.
Q.3 接触 does not only mean physical contact. An example of a non-physical contact may be 眼神接触 which means that their eyes meet each other. But for physical contact or 肢体接触, the range of physical contact may differ from just a slight accidental touch to rugby or american football style of clashing against each other.
Q.4 沟通 is comparatively the least formal way of saying communicate out of all the other words, but it also means the conveying of one message to another. It could be divided as 语言沟通 or 肢体沟通 in which one means the conveyance of message through words and the other is conveying through body language. I hope that this could sort of answer your question as there is a slight difference to all the words but they still mean communicate in English.
交际 as you have said, is mainly in a social manner where you would chat casually with friends or could even be used when meeting new friends.
For 打交道, it is not used in a "you" scenario, it could be used when countries make dealings which one another or when dealing with high government or company officials.
I hope I answered all of your questions and good luck in learning Chinese!