Can a molecule contain a linear series of conjugated pi bonds twisted through 180 degrees?
The material you imagine has been a target for Natta, best known together with Ziegler for they work on isotactic polymers. 

Natta was interested in what you describe - polyacetylene - as the extended conjugation would have likely led to unusual electronic properties. 

Calculations and models were around theoretically, before and in  parallel to the actual synthesis. 

The material prepared by Natta was however insoluble and basically no work up was possible.  This is not totally surprising if we look at the structure. 

After decades, the fortuitous use of thousand time the amount of catalyst led to the formation of nice shiny films of polyacetylene. Still insoluble but so different from commons residues to attract sufficient attention to cause the gathering of brilliant scientists and finally the emergence of an entire field, that of conducting (or better, conjugated) polymers and so called plastic electronics.

https://www.nobelprize.org/nobel_prizes/chemistry/laureates/2000/

https://en.m.wikipedia.org/wiki/Polyacetylene

This is to answer the final part of the question. 

Concernig the first part, if I understand correctly your picture, I can tentatively answer in an apparently convoluted way. 

As far a chain of alternating single and double bonds is considered, and without accounting for delocalisation, we have basically free rotation around the single bound. As such, any statistical coil can be obtained.

Conversely, if one imagine a "metal like" structure with full electron delocalisation, the chain is than stiff.

The reality is somewhere in the middle. Real conjugated polymers should be regarded as delocalized pi systems (which at the end is the reason of the huge interest on them) inerhently not much flexible and not very soluble. 

Still, a long chain is susceptible to adopt a variety of elical or coiled conformations as a result of collective little departures from planarity. Ideally, you can turn very rigid rod on themselves providing they are long enough.

One cannot apply Flory theory as in polyethylene, but adaptations exist to account for rigidity due to order bond > 1 in conjugated polymers. 

Also let me note that independent of the extent of delocalisation (also known as effective conjugation) a linear series of sp2 hybridized C atoms can close to a ring - no need for examples here).



Additional notes:

1- Solubility in common solvents is a major challenge towards real  application of conjugated polymers. A common strategy is the side substitution with moderately long alkyl chains. This in generak results in improved solubility via an extra entropy term.

2- polyanilyne far preceded the interest on polyacetylene. This systems is more complicated and in a way interesting, but less as far organic electronics is in scope.

3*- a detailed and chemically exhaustive paper appeared well before the paper mentioned in the linked Nobel prize. It has been published in the Journal of the  Australian chemical society and it details the preparation and properties of polythiophene.

*(Making products and selling them are two different tasks. This is the lecture I got)