"...there are only 2 black wires, and 1 copper wire (ground). The instructions say i have to hook up the white wire to a set of white wires in the junction box. but there are no white wires in the junction box." I'm a bit confused by this description. Did you mean that there are 2 black wires, 1 copper wire and 1 white wire?. That's what would make sense. Assuming that's the case, here's the deal: There are a couple of ways switches can be wired. One way is that the power feed first enters the switch box with both a hot (black) and neutral (white). The neutral is simply passed on to the fixture (i.e., the incoming white is connected to the outgoing white). The incoming hot connects to the switch and an outgoing hot goes to the fixture. (So, two wires in, two wire out, right?) A second way is that the power feed goes first to the fixture. The white (neutral) connects directly to the fixture and the incoming hot goes to switch along with what we'll call a "return" wire. This is known as a "switch loop" and it has no neutral. One wire is "always hot" (the feed) and the other is the "switched hot" because it only gets power when the switch is on. [There are a couple of other ways to wire a switch, but these two are the most common.] A simple switch does not consume power itself. It only connects the two wires then it is closed ("on") and disconnects the two when it is open ("off"). The problem arises when a more complicated switch needs power for something. For example, a switch with a pilot light or night light consumes power itself and so it needs a neutral. Ditto with some timers. You cannot wire a switch that is itself a power consumer to a simple switch loop, which is apparently what you have. So, there are two possible solutions. First is to do some rewiring to bring a neutral into the box. One way to do this is replace the switch loop cable (two wires) with a three-wire cable, one of which will connect to the neutral at the fixture. Since you're not familiar with wiring, I won't elaborate on how to do this, or other ways this can be accomplished. You'd probably need an electrician. A second possible solution is to find an old-fashion mechanical timer that doesn't use electrical power itself. (It's basically a switch with a "wind up" spring.) This assumes, of course, that your intended purpose is just to let the fan run for a while and then switch off by itself. If your objectives are more sophisticated than that, I'm afraid you're SOL. Good luck.

You can get an axial fan that goes in the duct - then all you need to do is cut a hole in the ceiling tile - for the grill that attaches to the duct.

The installation of the bathroom exhaust is a big problem, but not your only problem. The overarching issue is that your kitchen exhaust is from the ceiling rather than a hood over the range/cooktop. Trying to work with an existing ceiling exhaust, even if you replace it with a correct ceiling exhaust, well, it's still a ceiling exhaust. Take this opportunity of your kitchen reno to improve this arrangement. Lots of problems with a ceiling exhaust, most notably: (1) too much opportunity for grease and smoke to escape to the rest of kitchen, (2) difficult to clean at the ceiling, (3) not enough cfms to pull the smoke and grease from such a distance, and (4) no canopy shape (as a hood would have) to corral the grease and smoke. Take the opportunity of your reno to install a correct hood exhaust. Edited to add: I just noticed you said you are in a condo. If you are stuck with that exhaust location, you can still install an "island hood" in a chimney style that meets the ceiling where the duct is located. Make sure the hood is in a canopy shape, and is 6" wider than the width of the cooktop.

We have this one, it’s on a sloped ceiling https://www.amazon.com/Broan-NuTone-164-Infrared-Two-Bulb-Ceiling/dp/B000VTSN0E