1) Yes, Welkins, it's a gerund noun phrase, the direct object of finish.
2) No, not in this case. Now, as to your sentence, it doesn't work and here's why:
Now you've got a present participle phrase, which is used in a sentence to show two simultaneous actions. That's why the sentence doesn't work. You can't be making sandwiches at the same time that you're passing them to other people. It doesn't make sense.
To fix the sentence, we need to use Having + past participle, which shows consecutive actions:
Having made the sandwiches, he passed them to his wife and children.
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