Pressure loss in an infinite pipe I feel lost in this problem. I know pressure loss occurs in a pipe flow due to friction.
Say given a constant diameter pipe, if we keep lengthening the pipe horizontally, and static pressure will decrease in the downstream.
$Av=\text{constant}$ requires velocity $v$ be constant. [mass conservation]


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*Then we must be able to reach to a point where pressure $p$ drops to zero, is that correct? Then what happens after that? Will pressure keep dropping to negative values?

*If not, will there be a minimum pressure for which a given fluid and flow scenario pressure will sustain at?

*If so, energy conservation requires that velocity has to drop as well, so the flow will come to a stop eventually? Do we have such kind of "velocity loss" thing? If there is, what is the relationship between this "pressure loss" and "velocity loss"?

*If there is such thing as "velocity loss", then what about the mass conservation? Should we then need to discard the "incompressible" assumption?

 A: What comes first? Pressure or flow? 
Without potential, a pressure differential, there is no flow. It's pressure that needs to be there first. So rather than ask what pressure drop you get, it's better to ask what flow you get from applied pressure. 
We can never measure pressure drop across an infinite pipe because we can never reach the infinite point. But just considering a very long pipe, there is allot of resistance. So to move fluid you need lots of inlet pressure over what pressure you have at the outlet. And just increase the pipe a little longer, and more pressure is required. At some point you cannot provide the energy needed - even before you reach infinity. You will have a huge (static) pressure differential but zero flow.
Resistive losses in a pipe only occur when there is velocity. But there is something missing in your model - inertance of the fluid. The potential energy you supply at the inlet must provide sufficient force to first accelerate the fluid, then enough pressure to maintain the flow against the resistance.
Just by turning the question around with regards to cause and effect you see all is consistent with what's expected. Pressure doesn't happen after flow. It was there to begin with, however changes when flow is activated.
