Sure, but carefully. By painstakingly following instructions, we finally accomplished our goal. -This is a clause correctly attached to the sentence. "By painstakingly following instructions" is not a complete sentence, because there is no subject or main verb. By the light of the moon, we found our way home. By the old oak tree stood a beautiful deer. By bedtime, he was exhausted. These are also correct, but usually the sentence created is a little complex, and mistakes could be made easily.

I'm not sure if there's a rule, but here's a few sentence structures: By + [time], [statement]. = The phrase will have been completed by the designated time Ex: By 9:00, it was already too late. By nighttime, we will have arrived home. By now, he should be here. By then, it will have happened. By + [adj. verb phrase], [statement]. = The [phrase] occurred, because the [adj. verb phrase] did. Ex: By running frequently, you can get fit quickly. By carefully and intelligently managing our assets, we improved our finances by over 60%. By doing something, something far more dramatic happened. By trying to think of examples using the word 'by', I completely exhausted my imagination. By the way, + [statement]. = Used when the topic of the statement is not directly relevant to the current conversation. The statement is often a question. This form is used to partially or completely change the conversation - it makes it a lot smoother. Ex: A: That steak looks delicious. B: By the way, did you see the price of steaks this week? A: This weather is very good. B: By the way, did you hear about...?

Yes, for example: By the time we get to the cinema, the film will already have started

By the way, that is an interesting question. I'm not too sure if my previous opening sentence is a valid one.