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LingakuLingaku[verb て-form]、[sentence] [verb て-form] + [verb] [verb ない-form] 〜ない → 〜なくて、[sentence] [verb ます-stem]、[sentence]
The て-form joins two clauses that are not steps in a sequence: the first gives the cause of the second, the way it is done, or a fact that stands beside it. The verb goes into the て-form, and a negative becomes 〜なくて. Only the last verb carries tense and politeness.
Nothing in the て-form says which link is meant; the verbs decide. Walking is a way of going, so あるいて歩いていく行く is "go on foot". Waking up is not a way of washing, so あさおき朝起きて、かお顔をあらった洗った is one action after the other. That "and then" use, and how to build the て-form, are separate points.
The first clause says what happened, the second what came of it, closer to "and so" than to "because". The second clause reports an event, what you can or can't do, or how you feel.
Very often the second clause is a feeling: you are glad, sorry, frustrated or surprised because of what the て-clause says.
A て-form cause cannot lead into a request, an order, an invitation or something you mean to do. Those need から or ので.
A て-verb straight before the main verb says how the action is done, or what state you are in while you do it. The two happen at the same time rather than one after the other.
For "without doing", the negative is ないで, not なくて.
The choice follows the word, not the meaning. A verb goes into the て-form; a noun takes the particle で for the tool, the vehicle or the language. で never follows a verb.
Two facts that hold at the same time, about different people or things, can be joined with て. Neither comes first, and each side often has its own は.
These go beyond cause, manner and side by side.
Sometimes the second clause goes against what the first would lead you to expect. Context alone carries the "yet".
In formal writing the ます-stem takes the place of the て-form, in any of these uses. In speech, use the て-form.