Lake Cook Exteriors will be completing a siding coverall which includes siding, soffit and fascia, gutters, windows and doors. Does it matter what order the different types of installations should be completed?

The work can be done in any order. Ideally, a roof-tear off, including gutter removal, would be finished and then the gutters installed afterwards.  New windows and doors can be installed before, after or at the same time the siding is installed.  Soffit and fascia can be installed at anytime.