Electronic – Is shielding the entire circuit with a faraday cage a good idea

A simple radar should give you some idea.

If it's inside a small enough cage, possibly a near-field effect-on-a-tuned-circuit type of system would work, instead of needing a pulse-type radar.

This may not be a perfect indicator - it could be hard to tell the difference between being in the midst of a number of conductive planes, and being inside a complete faraday cage. And a cage with a perfectly absorbing inner liner would presumably look like free space.

Assuming you are somewhere in an intact industrial society, detecting existing external sources (broadcast radio, mobile phone networks, etc) is probably more practical. Even absent that, there are cosmic radio sources you should be able to see if something isn't in the way (absurdly ionized atmosphere?)

Bear in mind that what qualifies as a Farday cage is a function of frequency; you would need to use test frequencies comparable to the bands for which you need to know the answer.

There are several factors to consider when assembling RF amplifiers in VHF range on a pcb. No claim of being exhaustive, just some thoughts for discussion.
1) amplifiers need supply that in turn need decoupling: this is the path #1 for undesired effects; in an old project I was distributing power to several modules using large traces from a common +28V and decoupling with pi C-L-C circuit; traces and not a single plane to avoid common impedance coupling through V+. For large power, and large current since supply voltage is never above some tenths of Volts, you need to design for heat dissipation in your traces (e.g. 35/70 micron of copper, outer traces for power, ...).
2) Spread around capacitors to improve supply impedance at VHF; not such a high frequency to require distributed capacitance, e.g. through pcb itself.
3) useless to say, a good ground plane gives capacitors a good reference and is beneficial to reduce ringing
4) you say that A2 and A3(1) can be bypassed by 0ohm resistors: consider that all these modules shall be fed with transmission lines built for 50 ohm (or whatever it is) characteristic impedance; those 0 ohm are a discontinuity and a cause of mismatch for return loss and backward power. Better if you use RF relays or you manually reconfigure a coax line.
5) if amplifiers are modules in metallic housing, probably they are already well shielded; if they are components/hybrids, then shielding is necessary as you propose, but ... first, the shield might electromagnetically load and change module response (not dramatically), and, second, considering your 110 dB gain, keep amplifiers and filters separated, using metallic separators, otherwise noise pickup between output and input and oscillations are likely to occur. What is normally done when using some kind of shielding enclosure is to use coax capacitors for decoupling when passing through the wall.
6) consider distributing signals as striplines inside the power layers, rather than microstrips, to reduce stray coupling and radiation; frequency is low enough so that both work from a purely RF viewpoint.