How much do relays effect the sound?

Most people feel relays are better than a manual switch, but of course not all switches are created equal.  You won't get the same potential for crosstalk or noise with a relay that you will with a manual switch.  Also, you can use a separate transformer and regulator to provide the power that controls them which will isolate it from the signal path.

The downside of relays is that they make a clicking noise in the chassis which some people don't like.  Also there are latching and non-latching types.  Latching types only use power when a signal is sent to them, but they require a microcontroller.  Non-latching are more common, and require power constantly flowing through them to maintain the contact.  There should be no difference in sound quality between either one, especially if you keep the power to the relays on a separate circuit.

In my opinion, input selection is best done by relays.  With attenuation, you typically get more increments when using a relay-based attenuator than you will with a manual stepped attenuator.  Stepped attenuators and relay-based attenuators both use resistors, so ultimately it depends on the quality of the resistor being used.

Another advantage to using a relay is that it's completely sealed - something that doesn't seem to be hugely common with rotary switches, but may help with reliability in the long term as the contacts won't be exposed to quite as much grime.

Not all relays are sealed. The contacts can all be different too. I've seen copper contacts, silver contact, nickel plated something, and can't remember what else. Relays are cool though. I like them.

The relays selected for the δ1 attenuator and δ2 input/output selector are sealed, with silver twin crossbar contact mechanism and gold-alloy contacts.

That sounds like one nice relay. I found some a while back which has various gases in them also. Not sure what ever came of them though.