Yes your pass is valid on this route as well as nearly all trains in the UK.

Trains run by LNER to London Kings Cross are the quickest and most frequent option, journey from 4h20 - 4h40 and they run every 30 minutes for much of the day.

Trains from Edinburgh run by Avanti to London Euston are considerably slower (their fast trains to London go from Glasgow) also a third operator Lumo run a handful of trains per day, I am not certain as to the validity of passes on them so best avoided.

Reservations are not compulsory on these or any other train in the UK (except sleeper trains and Eurostar), however if you want you can reserve seats for free at any UK ticket office. If you want to reserve online you can also do that for free, for LNER only use their website https://www.lner.co.uk/travel-information/make-a-reservation/ , you will need to create an account. For reservations for all UK companies sign up for an account at www.gwr.com 

Make sure you reserve a seat in the correct class, if you only have 2nd (standard) class do not reserve in 1st, it won’t be valid.