There is nothing to be gained from activating a pass far in advance and it makes it harder if you change phones or need to cancel and get a refund.

A small number of reservation booking sites look for pass cover numbers, these are not present on mobile passes, even activated, they can be generated here:

As well as the eurail site there are many other sources of reservations that can be used, in many cases they are a better alternative due to cost or reliability.

London - Paris can be booked here https://www.b-europe.com/EN/Booking/Pass#TravelWish

A list of alternatives with guides is on this thread

Reservations are not linked or displayed on your app/pass, they are separate and remain so. As long as the reservation is for the train you are using it is valid with your pass which is your ticket.

You still have to add the individual trains to the pass, this as well as activating the pass and pass days can be done close to when you travel.

Activating pass and pass days requires internet connection so keep that in mind, adding or altering trains as you travel does not.

For Eurostar bookings this is the best source, you do not need a pass number but it will probably need passenger names. 

 https://www.b-europe.com/EN/Booking/Pass#TravelWish

You do need a Pass Cover Number for making reservation at b-europe @Yorkie 

The link for getting the PCN is provided by @Al_G above.

Many apologies - I was thinking of when you look for availability. I assume you mean once you go to the basket. It is a few months since I used it that far but have looked a few times for availability where it only asks what type and class of pass you have.