싶다 is a auxillary adjective(보조 형용사). Auxillary verbs/adjectives can never be used on its own. They can only be used with a preceding main verb/adjective(본용언). Sometimes, an auxillary verb/adjective has the same form as a regular verb/adjective, e.g. 보다, and their usage and meaning differ significantly. But 싶다 is not one of them.
The preceding main verb must be conjugated, as with all verbs. Verbs are conjugated with verb endings(어미), and there are lots of them. Each different verb endings used on the main verb will modify the nuance/tone or meaning of the word slightly.
As you might have guessed by now, -고 is a verb ending. The dictionary lists 6 definitions for this word(I've highlighted the 2 definitions relevant to your examples):
두 가지 이상의 사실을 대등하게 벌여 놓는 연결 어미. Connecting verb ending treating multiple clauses equally.
앞뒤 절의 두 사실 간에 계기적인 관계가 있음을 나타내는 연결 어미. Verb ending indicating that the two facts in the preceding and the following clauses are non-coincidentally related.
- 오빠는 나에게 얼른 눈짓을 하고는 나가 버렸다.
앞 절의 동작이 이루어진 그대로 지속되는 가운데 뒤 절의 동작이 일어남을 나타내는 연결 어미. Connecting verb ending indicating that the action in the following clause happen while the action in the preceding clause is ongoing.
서로 뜻이 대립되는 말을 벌여 놓는 연결 어미. ‘하다1’의 선행 형식이 되게 하거나 명사적으로 쓰이도록 한다. Connecting verb ending which connects two opposing words.
- 누가 우등생이었고 누가 열등생이었고 간에 서로 친하게 지내도록 하여라.
형용사 어간을 반복하여 그 뜻을 강조하는 연결 어미. 흔히 ‘-고 -은’ 구성으로 쓴다. Connecting verb ending to emphasize to meaning of an adjective by repeating the stem of it. Commonly used as '-고 -은' structure.
본용언에 붙는 연결 어미. Connecting verb ending attached to a main verb/adjective.
Your first two examples are related to the 2nd definition(the one I highlighted first). Your third example and -고 싶다 construction is related to the last(6th) definition.
It seems that, according to the dictionary, -고 doesn't add any meaning when used in a "[main verb] 고 [auxillary verb]" type constructs. In other words, using -고 as the main verb's verb ending is the most generic usage of the auxillary verb. It just connects the main verb and the auxillary verb.
Note 싶다 has other usages, depending on the verb ending used on the main verb. Such as -면 싶다, which indicates that the statement is a small wish of the speaker.
- 오늘은 일찍 잤으면 싶다. I hope I sleep early tonight
For other uses of 싶다, check the dictionary.