The closet might be a good idea, but ducting from the house AC system will only work when part of the year - that closet will need cooling year-round, and will not appreciate it when you switch the house over from cooling to heating. Ventilation is your best bet, but if you have such significant dust problems you may want to arrange your equipment rack or closet so that you have a controlled intake point that you can put a filter on. If you moved everything to the closet, you could add a door grille with a filter on the back as the air intake and put a ventilation fan at the top. If there's an attic above, it would be a convenient place to discharge heat to.

Have you measured the air temperature inside the cabinet that you are currently using? It might be possible to make some changes to the rear of the rack to allow the front door to stay closed and keep the air temperature inside the cabinet down to a manageable level and introduce a filter of some sort to help cut down on the dust.

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