I think an FM Yagi is not going to work well for you, it won't have the directivity or the impedance match you expect at 65MHz. You're better off making a regular dipole out of 2 pieces of wire, or even better making a small loop of wire as a nearfield probe. Of course, if all you want is a relative measurement, any random piece of wire at a suitable distance will probably do as long as you don't move anything between measurements.
As for the yagi, baluns can have electrical isolation (or not, depending on their type) and impedance transformation. The characteristic of a dipole at its design frequency is 72 ohms, but when you put parasitic elements near it to form a yagi the impedance drops. Often, designers then move to a "loop-fed" yagi (the loop is actually a folded dipole at 300 ohms), then when parasitic elements drop the impedance you can end up closer to 72 or 75 ohms again and have less work to do in the balun.
The other role of a balun is to decouple the feedline from the antenna, and that's a whole different rabbithole because of skin effect. Coaxial cable should be thought of as three conductors - the outer surface of the centre conductor, the inner surface of the shield, and the outer surface of the shield. The outer surface of the shield can certainly become part of your antenna if you're not careful.
The parasitic elements of the yagi can either be electrically connected to the boom or isolated. The design changes slightly, but a DC short circuit is very different to what happens at radio frequencies.
Earthing is easy to do for DC (or for lightning protection) but much more challenging at RF. The key aspect as it matters to you is that any DC electrical connection is also an opportunity to inject noise into your measurement.