[Eliot Ness]

It is difficult to answer questions unless you provide a link to the schematic you're using. This is the schematic I found:

http://www.nostalgiaair.org/PagesByModel...013151.pdf

Yes, the IF transformer should have continuity on both sides. Sometimes old solder joints are hard to measure and you need to use a sharp probe to get an accurate reading. I use something like this:

http://www.newark.com/tenma/72-9320/test...P=KNC-GPLA

But be careful, you don't want to poke so hard that you damage the very little wires going to the lugs.

The capacitor block is #7 on the schematic I referred to earlier, and yes you should replace them....you can use the typical yellow 630V film capacitors that are available from most vendors. There is a member here (Buzz) who has posted Youtube links and one of his later ones shows him taking apart a can like that to replace the capacitors.

I'd also start looking for another speaker if I was serious about restoring this radio. I didn't notice wire nuts in your picture, but someone has been in there and used some modern wire for their repair. And if they crammed a modern speaker in there I'd be suspect of their other repairs.