Must a sequence be well-founded? 
Must a sequence be well-founded?

Is $\Bbb Z=(\ldots-1,0,1,2,\ldots)$ a sequence?
Conventionally we think of $\Bbb N=(0,1,2,3,\ldots)$ as a sequence, but what about if it has no starting value?
Obviously we can reorder any countable set $X$ into a well-founded sequence e.g. by an injection $f:X\to \Bbb N$ but what about in its raw form, do we call $\Bbb Z$ a sequence?

It depends on context.
In most contexts, a sequence is understood by default to mean a $\mathbb{N}$-indexed sequence.  So you can talk about e.g. a $\mathbb{Z}$-indexed sequence, but if someone says just “a sequence” without specifying the index set, they should be assumed to be talking about $\mathbb{N}$-indexed sequences.
In some areas of mathematics, the convention is different.  In set theory, for instance, sequences indexed by arbitrary ordinals are very commonly used, and so “a sequence” may be used to mean a sequence indexed by some ordinal, even if that’s not explicitly specified.
More generally: if you do specify the domain, how general can it be — arbitrary total order, well-order, poset, …?  Well, there’s no standard fixed definition of sequence to restrict this; a sequence is just a function, and the circumstances where one calls a function a sequence are just a matter of field-specific convention.  $\mathbb{Z}$-indexed sequences are certainly fairly commonly used; I wouldn’t be surprised to hear sequence used for functions on arbitrary total orders.  I’d be slightly surprised to hear sequence used for posets, and very surprised to hear it used for functions on an arbitrary set with no specified order at all.
