Do toxin producing strains of E. coli constantly produce toxin or only under certain conditions?
1) That depends on the fact, if the protein in the bacterium is regulated somehow or it is constantly expressed. That could be different for different toxins and even for the same toxins in different strains, so there is no  plain answer for all strains and toxins.

2) That, I guess, also depends on the toxin. There are some that are basically toxins, but do not make people sick. Sometimes, the toxin is produced by the bacterium, but never secreted. Sometimes it is secreted, but has no influence on the host. Sometimes the host does react to a toxin. But the human immune system can also react to the bacteria before they start producing toxin (if it is somehow regulated) or before the toxin concentration secreted by the bacteria starts to be harmful.

In the case of E. coli O157:H7, this paper states it produces two different Shigatoxins (that were supposedly causing its pathogeny), but also some other toxins/toxin-like proteins. 

I have studied some E.coli genomes previously, some non-pathogenic had several toxin-related genes, some pathogenic had actually none at all (at least none that could be annotated).