I found this transformer and I am not sure if it is working. I am confused with the multimeter readings. For example: when I am using the buzzer option to check the connectivity it buzzes when I connect any of the 3 top ones (16ohms and 32ohms), but when I try to see the connection between the 665ohms it doesn't buzz, but it does show ohms when I use the 2K ohms option. Is this how it's supposed to work?
What you have here is a center tapped transformer. The primary winding in this case is the 665 ohm one, and the secondary the 32 ohm one which is split into two windings, so each one giving 16 ohms. I don't know what the rating of this transformer is, but if we assume a 230V/12-0-12V transformer you would get voltage measurements between the windings as shown in the picture. You can find more info about these kind of transformers here: https://www.electronics-tutorials.ws/transformer/multiple-winding-transformers.html
But to answer your question, for this type of transformer all of your readings seem normal for it to be in working condition.
(Image source: 12-0-12 Center Tapped Transformer - Components 101)
(Image source: The Center-tap Transformer - Electronics Tutorials)
The readings look reasonable.
You will need to arrange a source of AC to test the transformer further. A known good mains to low voltage transformer is your best source. Many light strings use 12, 16 or 24V AC and transformers are sometimes found for sale at thrift shops.
If you have an old car or motorbike battery charger you should be able to connect to the transformer outputs before the rectifier, solder secure connections and lead them out safely and use with the case closed to keep the mains isolated.
A random bicycle dynamo or standard (permanent magnet) stepper motor chucked to a variable speed electric hand drill will give you some AC at a usually safe voltage and reasonable frequency. Make a little jig to hold it together and then hook it up to the high resistance winding and measure all the voltages and calculate the turns ratio.
If it is a 115V primary it will die if you give it 230V on the primary side.