When does it feel more natural to have articles in names?

There is no rule. Tradition, the sound, history, and some other factors are what matters.

Consider:

Bath, a city in England, was actually the Bath up until the Nineteenth Century, when the article was dropped.

Facebook was initially the Facebook.

There's some logic behind the New York borough called The Bronx: it was named after the family of Jonas Bronck, i.e. the Broncks. However, no one is quite sure why the Hague is called the Hague, and there are also the Vatican and the Stonehenge.

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You need to use the definite article with these:

the Royal Albert Hall, the John Hancock Center, the Kennedy Center of the Performing Arts

But not with these:

Carnegie Hall, City Hall, Avery Fisher Hall, Radio City Music Hall, Rockefeller Center, Lincoln Center for the Performing Arts, Epcot Center, Canterbury Cathedral, Westminster Abbey.

Also, compare:

London Bridge, yes, but: the Brooklyn Bridge.

"I work for BBC. I got my job there after I left Post Office." Not ok.

"I work for the ITV. I got my job there after I left the DHL." Not ok.

It has to be the BBC and the Post Office. It can never be the ITV and the DHL. Nobody knows why.

The possessive is also tricky. You can shop at Tesco's although the shop is named 'Tesco'. You could never shop at Ikea's. Again, nobody knows why, or, if they do, they aren't telling.