Closing off ALL outside openings should keep the crickets out, but whether they lay eggs in crawl spaces that could hatch out another generation in that environment (under the plastic) I do not know - ask pest control man. And whether they can chaw through the plastic to get out.
Good thought about the radon issue, but if the plastic liner is being properly physically clamped to the foundation wall, that should take care of the issue, I would think. That is standard practice for crawlspace radon control, though running a stack pipe to maintain a negative pressure under the plastic would be a standard procedure.
However, with that situation you might want to consider raking out any rocks and pebbles in the area and having them place a 1/2 to 1" layer of DRY (to prevent mildew smell) masons sand first, so workmen placing the insulation board and any future workmen do not puncture the plastic sheeting. That is a good practice anyway, because of the likelihood HVAC or plumber or electrical contractors might have to get in there in the future, and of course occasional pest control inspections, and laying on the board could cause sheeting punctures. Another alternative which might work as well and be cheaper if low headroom, is a thick non-woven geotextile filter fabric as a cushion, after raking all larger particles out.
Good Luckk with the project, and may you enjoy many years of lower utility bills.