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Question is not very clear about the context of word 'sequence' here.

If I am to select 4 numbers out of four and arrange them in order then there are 4!*8C4 = 1680 different sequences possible.

If the word sequence refers to some arithmetic sequence or geometric sequence, then counting is going to change for sure.

Q: How many possible sequences of four numbers are there from 1 to 8?

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There are infinitely many possible number sequences, and infinitely many numbers which can appear in those sequences. Any and every number can appear in a number sequence.

Three numbers can be arranged in 27 different sequences if repetition is allowed, and in 6 different sequences if it's not.

There are infinitely many possible answers. One possible quartet is (1,1,1,235)

625 if numbers can have leading 0s, 500 otherwise.

There are many possible quartets: for example, 1,1,2 and 25.

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more than 1

There are infinitely many possible number sequences, and infinitely many numbers which can appear in those sequences. Any and every number can appear in a number sequence.

Three numbers can be arranged in 27 different sequences if repetition is allowed, and in 6 different sequences if it's not.

There are infinitely many possible answers. One possible quartet is (1,1,1,235)

625 if numbers can have leading 0s, 500 otherwise.

There are infinitely many possible answers. For example 1x1x1x196 10x10x10x0.196

There are many possible quartets: for example, 1,1,2 and 25.

There are infinitely many numbers between them. Also there are infinitely many sequences that wil have two numbers between 1000 and 1.

You have 2 choices for 20 times, so the number of sequences is 220 or 1048576.

There are infinitely many possible answers, particularly since the operations are not defined.One possible answer is 519.653 + 0.1 - 0.1 + 0.001

With the constraints you have listed, there are only 6 possible phone numbers:202-1178202-1187202-1718202-1781202-1817202-1871The first four digits are fixed, and there are six possible permutations of three different digits.

There are many possible answers. One quartet is -1, -1, 1, 315