A coach has a reasonable likelihood of being cheaper than a train, though a train will probably be the more pleasant option. Our roads are often congested so a train will usually be faster, and you'll have more space to move about, larger toilets, a place to buy snacks and light meals, etc.
London Underground is quite simple. If you're using a paper ticket it's just one charge no matter when you travel. If you're using a contactless card there are separate charges for peak time and off-peak but it'll charge you the correct one automatically depending on when you tap in. Depending on what ticket type you go for (see below) it may be worthwhile to buy a ticket that combines London Underground and mainline rail travel, in which case you won't need to worry at all about what time you catch the Underground train.
You basically have two options for ticketing: buy what's known as a "walk-up" ticket, which will be flexible and allow you to travel on any train (with some defined rules requiring you to avoid the peak hours, most likely), or buy an "advance" ticket which ties you to a specific train. A walk-up ticket is so named because you can walk up to a station on the day of travel and buy it. You can also buy them in advance online if you want to reserve a seat (this is not usually mandatory on UK trains, though it is right now due to COVID). They have names like "Anytime Single", "Off-peak Single", and "Super Off-peak Single" (note that as a result of a recent pilot scheme return tickets are no longer available for this route; buy two singles); most online train booking websites now do a decent job of making it clear you're buying a flexible ticket. Advance tickets can only be bought in advance (hence the name), and tie you to a specific train through the means of a mandatory seat reservation (without which the ticket is invalid), but can often be a lot cheaper than the flexible alternatives.
- Yes, long-distance trains on that line all start their journeys at King's Cross, so you should have plenty of time to walk along the platform to find your coach and seat.
It's worth pointing out that right now there's major engineering works at King's Cross and so a reduced service is being run, so trains are likely to be busier than usual (though in practice I don't know how busy that is, due to the reduced travel from COVID). Depending on when you are travelling this might affect you, though I'd hope that the timetables online have by now been updated so any train you book on should actually exist... in the event that the train you are booked on doesn't exist and you have an advance ticket, you'll be entitled to travel on the next one by the same operator (LNER in this case).
