When you name the river, you capitalise river, so it would be:
River Jordan
I swam in the River Jordan
but without the name:
I swam in the river.
The presence of 'the' is less relevant than the presence of the name of the feature. There are similar conventions with other natural features, for example mountains, e.g. Mount Everest.
Capitalization (especially of titles) is a tricky business in English. There is no hard-and-fast rule about capitalization. In fact, many of the major style guides disagree on exactly what should or shouldn't be capitalized. However, I'll give some general guidelines and let you make up your own mind:
Titles in direct address
In all major style guides, when a title is used as a name in direct address of that individual, it is capitalized. For example: "That was very kind of you Mom." "I love you Dad." "Stop being so good to me, Master." "So, I have to have surgery, Doctor?"
Titles used in direct reference
When referring to a specific individual with their title, the title should be capitalized. For example: "I was excited when President Barrack Obama walked in the room." "My favorite comic book character is Mr. Tony Stark." "It took a while, but we finally got in to see Doctor John Smith."
However, (somewhat less commonly) some prefer that occupational titles not be capitalized, unless abbreviated. For example, "Dr. John Smith" and "doctor John Smith".
Titles used in indirect reference
Most style guides prefer titles used in reference to be lowercase. For example, "I love my mother." "The president gave a great speech." "My doctor did a great job with my surgery."
However, some titles are exempted from this. God (when referring to the Judeo-Christian god) is always capitalized. The King and the Queen are often capitalized. (For example, "The Queen has quite a way with words.") When referring to a specific office, there is disagreement about what should or should not be capitalized. Chicago and Wikipedia prefer "the president of the United States", while NY Times prefers "the President of the United States". However, everyone prefers "King of France" to "king of France".
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Whether to capitalise/capitalize the name of an empire is a matter of style. British publications are inconsistent (or uneasy) about capitalising. The (UK) Guardian (centre-left) style guide gives 'British empire' as the expected style, whereas the Daily Telegraph (centre-right) guide has no mention of the term, but has plenty of articles with "British Empire" in them. The (American) Chicago Manual of Style is lukewarm and says that a writer risks...
Capitalization (Titles)
Many American academic and commercial style guides recommend capitalising, and the general rule seems to be to capitalise words derived from proper names. Webster's II New Riverside University Dictionary says:
Capitalization
Whichever style you choose to follow, be consistent, that is, if you capitalise 'British Empire', and you mention others, you should do the same for them, for example the Roman, Ottoman, Byzantine, Holy Roman Empires. Also to note: you would never capitalise a zone of influence which is sometimes called an 'empire' of a nation, but is not officially ruled by that nation, so that if you write about the 'American empire', you should lower-case 'empire'.