Is Trainline recommending you use London Underground to get from Heathrow to Liverpool Street? Long story short, Trainline is shit at showing underground journeys - I would ignore the arrival into Liverpool Street being at 9:30. Instead https://citymapper.com/london?lang=en will help you plan your journey between Heathrow and Liverpool Street properly with the journey times. Their mobile app will be useful on the day to help you navigate any delays etc.
This one? Looked very clean to me. The only place I could see it connecting to was the TfL site when it gets tube status information. There is also no privacy policy in the app, which there would need to be if they were collecting any information. Perhaps it was just someone's pet project?
I just checked and it's not available :/ it was Richmond Toys but I found this one: https://www.amazon.co.uk/London-Underground-Train-Toy-Model/dp/B09DDG4WPT/ref=sr_1_4?crid=23WVN52SXHJPM&keywords=london+underground+train&qid=1643448673&s=kids&sprefix=london+underground+train%2Ctoys%2C103&sr=1-4
Personally, if you're only visiting for a short time, I don't think you need any sort of discount card. The Railcard costs £30 and the ZIP costs £20, so it's not going to be worth getting unless you're here for a month and are travelling every day. You can use a normal Oyster card which you can buy at most stations, or you can just use a contactless credit or debit card if you have one.