Eurostar to Paris and Brussels.
About 60 miles, but you can't travel there directly from St Pancras, you would have to change trains.
You don't. You travel from Sheffield to London by ordinary trains.
Look at your Tube Map and travel to Kings Cross/St Pancras Station.
Yes, you can use the northern line.
Backpacking it, staying in hostels and using the slow trains for transportation makes for the cheapest travel around Europe.
You can find information regarding European trains on Rail Europe's website as well as Eurail's website. You could also speak with a travel agent to help you find and or plan specific routes through Europe.
Trains go from PARIS AUSTERLITZ station in 1h 30min
Vienna
Trains travel on rails ...
you can take heathrow express from airport to paddington station , then from there you can take a bus to London St pancras station and then train to dover, or you can use a taxi which can be cheaper if you travel with your family.
The main options are either flying, London has a big selection of airports to choose from, or travel by Eurostar. Eurostar operate from Ebbsfleet or St Pancras.
So that trains could travel between England and continental Europe.