Re: Valve covers
At least an EF80 has the mesh screen round the anode so it's less likely to be affected by extraneous variations in anode-to-the-rest-of-the-world capacitance. And if my memory's correct the AT5-style osc is an "electron-coupled" Vackar-type design where the valve's screen-grid acts as a 'virtual anode' to the oscillator itself, which should also give greater isolation from real-anode-capacitance fluctuations.
[Oscillators can be fickle beasts - in another thread I narrate the hassle of trying to stop the 'ant trim' control of my Trio 9R-59 acting as a VFO-shift. This uses a triode [half an ECC85] and the anode-to-earth capacitance definitely does have an influence on oscillator frequency in this case! From the factory it has a close-fitting screening can [a tube with a slot cut down the side to provide springiness] and while this is needed to stop the anode-to-world capacitance varying it does make the valve run rather hot]
Heater wiring - I'd say that you should earth any pins that are not-used and are also not part of any 'internal connection' to the valve. And decouple both heater-pins to earth with 0.01uF if you're using 'balanced' heater wiring rather than one side of the heaters being returned to chassis. The only reason I can think of for a separate earth to one end of the heater rather than commoning it with the cathode and suppressor-grid and mesh-screen is that there's a *vague* possibility of any common impedance in a shared earth [corrosion between earthing-tag and chassis?] resulting in a squib of heater-current AC being injected into the RF side.
[My AT5-alike transmitters always kept the heater wiring to each valve 'floating' above ground - though thoroughly RF-decoupled, because it made 6/12/24V supply-switching a lot easier]
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