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.