For sure I'd take those seleniums completely out of the circuit. Having them in parallel with diodes is a bad idea.
I'm always paranoid about damaging a CRT. Especially when it's a series filament wired set like this. If there's excessive current for some reason, there goes your CRT. So I like to take it out of the circuit and substiture for the filament with a 10 ohm 5+ watt resistor. It's normally 6.3 VAC @ 0.6A. That's where I got the 10 ohm value. You can just pull the plug off the CRT and stick the resistor in the filament holes. Those are on either side of the key:
http://www.bunkerofdoom.com/tubes/sy...S/10ABP4_1.JPG
I don't think these sets came with an antenna. I think the ones we've seen in photos are after market accessories.