King Siddharta, his wife Jasodhara, and other buddhas

Buddha's full life story is found in this book with his family tree. Buddha and his teaching by Narada

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One glorious day as he went out of the palace to the pleasure park to see the world outside, he came in direct contact with the stark realities of life. Within the narrow confines of the palace he saw only the rosy side of life, but the dark side, the common lot of mankind, was purposely veiled from him. What was mentally conceived, he, for the first time, vividly saw in reality. On his way to the park his observant eyes met the strange sights of a decrepit old man, a diseased person, a corpse and a dignified hermit.18 The first three sights convincingly proved to him, the inexorable nature of life, and the universal ailment of humanity. The fourth signified the means to overcome the ills of life and to attain calm and peace. These four unexpected sights served to increase the urge in him to loathe and renounce the world.

http://www.buddhanet.net/pdf_file/buddha-teachingsurw6.pdf

That seems to be several questions -- I'll try to answer them all but only briefly.

What happened to king Siddhartha after he left the palace after renouncing the duty of King?

He was never "king". He was son of a king -- or, Wikipedia says,

Shakya

Gautama Buddha (c. 6th to 4th centuries BCE), whose teachings became the foundations of Buddhism,[note 2] was the best-known Shakya. Known in his own lifetime as Siddhartha Gautama, the Buddha was the son of Śuddhodana, the elected leader of the Śākya Gaṇarājya.

After he left the palace he tried to gain enlightenment (liberation). He studied with several teachers, eventually succeeded when he was alone (thus becoming an "arahant"), decided to teach what he learned (which a Buddha does: teaches). This is summarised in Ariyapariyesana Sutta: The Noble Search (MN 26).

He then taught for the rest of his life, leading the "Sangha" (i.e. Buddhist monastics) and passing away at about the age 80.

How many days queen Yaśodharā remained in the palace with his son?

I don't know exactly. She eventually left the palace to join him, with the son who was named Rahula -- they both became monastics too, joined the Sangha -- so did Mahapajapati Gotami, who was Siddhārtha's step-mother and who asked that the Sangha be expanded to include nuns as well as monks.

Rahula eventually became an arahant too (which I think, from memory and without looking it up and so I might be wrong, was when he was a young man); and so did Yaśodharā.

A few suttas include the Buddha's teaching Rahula. I think this is an early one, -- Ambalatthika-rahulovada Sutta: Instructions to Rahula at Mango Stone (MN 61) -- and that sutta was when Rahula was age 7 or something like that. That implies that the princess and son didn't stay more than a few years in the palace after Siddhārtha left, possibly (I don't know) just long enough until Rahula was old enough to survive without nurses etc.

Wikipedia's biography of Yasodharā suggests it was 6 years before the Buddha returned to near where they were living, and it was then that they joined the Sangha.

A king is not so weak minded so that he he will give up kingdom. This is hard to believe he forsook every kind of luxury after seeing a Deadbody and a blind person or leper.what occured to him,many learned writers have written.

If you say so. Bhikkhu Bodhi for example sounded a little skeptical too when he wrote this essay -- Meeting the Divine Messengers -- you might like to read that.

After king siddhartha, there were other Buddhas which include Ananda and other Buddhas who took over preaching and practice.

People like Ananda are called "monks" (if and when they joined "the Sangha" i.e. the community of monks).

There are different degrees of liberation -- Four stages of enlightenment (Wikipedia) -- and though many other monks became Arahants through the Buddha's teaching-and-discipline, Ananda himself didn't reach that last stage (Arahant) until after the Buddha death (though Ananda too was teaching even before then).

None of them are called "Buddhas" though -- there is only one Buddha at a time (called 'Sammāsambuddha'), that Buddha then teaches others.

The monks teaching each other and practising wasn't "instead of" the Buddha teaching and practising (it was, "as well as").

The Buddha wasn't a king. The story is that it was prophesied that he'd become either a Buddha or a World-ruler, which was why his father tried to keep him isolated in a palace.

since king Siddhartha was king of a kingdom he did not live the life of a sage for a long period. This is obvious.

He never became king (not in that lifetime anyway) -- left the palace before he became king.

My question is what king Siddhartha Gautam and other renowned Buddhas were in the sight of new king and his ministry.

I don't know -- don't remember whether that's said anywhere, i.e. the relationship between the Buddha and the rulers of his former nation (including his father)?

One thing that is written is that that nation (the Shakyas) was destroyed by or conquered in a war.

Was King Siddhartha and Gautam Buddha was of same identity?

They're the same person (except, as I said, Siddhartha wasn't a king).

I'd tend to say "Siddhārtha" to refer to him before hist enlightenment, and the Buddha afterwards (or another translated title like "the Blessed One" -- bhagavant).

BUDDHADEB

I think I've never heard this word being used. Wikipedia says that was the name of a 20th century Indian politician.