We know about Bohr's model and his vague postulate challenging Rutherford regarding discrete orbits and not emitting electromagnetic waves during this.
Extending this idea to our solar system, does the Earth emit gravitational waves around its orbit of the Sun and if not why not, and if yes, will the Earth fall into the Sun?
Extending this idea to our solar system, does earth emit gravitational waves around its orbit to Sun[?]
Yes. In general, any matter distribution with a time-varying quadrupole (or higher-order) moment will emit. In practice, this means that you need to have a mass distribution that is not spherically symmetric, but the Earth-Sun system satisfies this.
[A]nd if yes,, will Earth fall on sun?
The rate at which the Earth-Sun system radiates power via gravitational waves is minuscule. If you look at p. 9 of these lecture notes, you'll find the equation for the average power emitted by two orbiting masses with semi-major axis $a$ and eccentricity $e = 0$ is $$ \langle P \rangle = - \frac{32}{5} \frac{G^4}{c^5} \frac{m_1^2 m_2^2 (m_1 + m_2)}{a^5} $$ For the Earth-Sun system, this works out to be about 196 watts of power.
The Earth has so much kinetic energy that this energy loss won't affect Earth's orbit by any perceptible amount over the lifetime of human civilization. You can also look lower down on page 9 for the equation for the lifetime of a circular binary system; the result is $$ \tau = \frac{5}{256} \frac{c^5 a_0^4}{G^3 m_1 m_2 (m_1 + m_2)} \approx 1.07 \times 10^{23} \text{ years}, $$ which is several billion times longer than the current age of the universe. I would say this is not a terribly pressing concern.
Presuming general relativity is correct, yes the Earth orbiting emits gravity waves. The intensity of this emission is quite low. The intensity of the gravity field the Earth creates is small. The speed the Earth moves around the Sun is small. So the rate of energy being carried away is very small. Thus the stability time of the Earth's orbit (w.r.t. this energy loss) is quite long. Other effects would almost certainly be larger. Tidal drag of the Sun on the Earth, for example. Probably little tugs from the gravity of other planets would be larger than the gravity wave effect. Probably the Earth's orbit will be stable enough for other dire things to happen first, such as the sun getting along in its life span, and drastically changing character. (I'm weaseling on that because I don't recall what is the expected fate of the Sun.)
