There is no limit on the size of the fragment that can be cloned into PiggyBac, some published reports have used over 120kb in successful transpositions.
If you vary the amount of Super Transposase included in the co-transfection, you can vary the amount of integration of the PiggyBac construct into the host cell genome. You can perform a dose-response curve along followed by qPCR for a region of the PiggyBac transposon that is not native to the host cells to measure integrated copy number.
For cloning and general propagation of minicircle parental plasmids, you can use any kind of competent E. coli. For induction of minicircle production, the ZYCY10P3S2T E. coli MUST be used because it is the only strain that has the engineered genome that causes the induction. Substitution of another E. coli strain for minicircle production will result in no minicircles being produced.
Most likely this is supercoiled plasmid DNA or nicked DNA or both. You need to do a restriction digest with 1-2 enzymes before running the minicircles and parental plasmids on a gel. This is the only way to determine the size of the produced minicircles and to determine whether the induction was successful.