You must take your permit with you if you go abroad while you have permission to stay in the United Kingdom. You may have to show the permit to an immigration officer at the border when you leave the United Kingdom.
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As a US permanent resident, you are allowed to freely travel outside of the US. To travel, you usually need your permanent resident card, a valid passport, and whatever visas are required by the country you intend to visit.
Spain Visa Requirements for U.S. Green Card HoldersWhile U.S. citizens currently don't need a travel visa or a waiver to visit Spain, you will need a travel visa if you're a green card holder originally from a non-Schengen country, as you are still considered a third-country national.
You always need a passport or other travel document for a journey abroad. This is also the case for a journey within the Schengen Area. Your passport or travel document must not have been issued more than 10 years ago.
Non-EEA citizens with UK Biometric Residence Permission (BRP) who wish to travel to Ireland may need a visa to do so, as their BRP has no bearing on their permission to travel to Ireland. You should also note that a Schengen visa is not valid for travel to Ireland.
Applying for a UK to USA Visa can be exhausting and demanding, but when you have a British passport, that's no issue as British citizens are under an American visa-waiver program. If you have a British passport you don't need a specific US Visa for UK citizens to enter the US, you only need an ESTA.
You need your passport. A residence card (from any country) is not a travel document (that can be used to travel to other countries). You will have to show your passport at French immigration control in London St Pancras. If you don't have it, you will not be allowed on the train.