Regarding grounding the antenna, per se: Antennas are not all made the same, and the receiving part of some antennas is not at all connected to the antenna mount, so grounding that is not the same as grounding the shield of the cable. Antenna grounding is meant to offer a path to ground for static charges in an attempt to bleed these charges off and lessen the likelihood of a lightning strike. If lightning is not typical in your area (as, for instance, for me in southern California), that ground is not required in practice though specified.

If touching the jacket of the cable improves your signal, you don't have a strong signal. Ditto touching the actual outer metal part of the F connector. And this CAN be improved by connecting to an earth ground.

The rental house has an earth ground or it is, legally, not habitable. For your purposes, though, that "less than $5.00" could be a royal pain if you tried to connect to the earth ground. However, you could verify that your power connections are made correctly, then connect to the power ground near your TV.

Take this advisedly, though -- I don't know what kind of electrical connectors are used in Canada. In the US, all three-prong outlets are supposed to be grounded, and the mounting screws of the outlet are connected to that ground, so there's a place for a ground for your antenna. Check this by getting an outlet tester and checking for proper wiring including ground. Be sure to test this with something plugged in and operating on the second outlet, if you have two together. DO NOT connect to the neutral of the outlet, and definitely not to the hot!