Your condo violates Code and is unsafe, and your HOA consists of idiots

First off -- your condo's existing setup, with a recirculating hood over a domestic gas appliance, is quite clearly a Code violation and a flagrant safety hazard, as IFGC 503.3 strictly prohibits the recirculation of gas appliance exhaust due to the unacceptable CO buildup that would otherwise occur:

503.3 Design and construction Venting systems shall be designed and constructed so as to convey all flue and vent gases to the outdoors.

Second, that ceiling exhaust in the laundry room likely was installed to service the dwelling unit -- you'd need to replace that ventilation system in order to reuse the existing exterior penetration for the range hood duct, and that leaves you back where you started.

However, if you can have an engineer analyze your ventilation situation as per Chapter 4 of the IMC and determine that the vent fan in the laundry room is indeed not necessary to provide Code-compliant ventilation to the dwelling unit as a whole, then you can disconnect the ductwork from the laundry room vent fan and reuse that penetration for the range hood, provided that the duct meets the conditions given in IMC 505.1 for domestic range hood ducts:

505.1 Domestic systems Where domestic range hoods and domestic appliances equipped with downdraft exhaust are located within dwelling units, such hoods and appliances shall discharge to the outdoors through sheet metal ducts constructed of galvanized steel, stainless steel, aluminum or copper. Such ducts shall have smooth inner walls, shall be air tight, shall be equipped with a backdraft damper, and shall be independent of all other exhaust systems.

The picture of your range hood indicates that it was designed to vent back into the kitchen area if desired. That is what those small louvered vents on the front are for. Most such units can be either vented through ducting in the traditional method (preferred method, IMHO) or vented through the vents on the front of the unit. They are called convertible duct units. When vented back into the room, a charcoal filter is supposed to be used along with (or in lieu of, depending on brand/model) the grease trapping metal mesh filter.

If you put your hand over the vents while the unit is operating, you will know if it has been converted to vent into the room because you will feel the exhausted air blowing out. if you don't feel air blowing out it is probably ducted.

You should not just assume it is not ducted, your picture shows a significant wall area above your cabinets that could be a soffit easily large enough to contain the ducting. Often a 4" x 10" rectangular duct is used (as opposed to a round duct) to save space, so there does not need to be a whole lot of room to run the duct. Inspect the rear of the cabinet above the unit to see if there are signs that a duct runs up behind, look at the outside walls of the home (in likely places) for an external duct hood. Or you could always drop the unit down a bit and take a peek.

As for your ceiling exhaust fan, I would take it out and thoroughly clean and de-grease the blade, motor, mounting hardware, and duct. If it still makes noise and does not run correctly you can buy a replacement motor for around $30-40 and it will run like new.