Do all the objects emit energy by heat? If it is so then why do they do so? Let's say we have an object at temperature 't' and its surroundings at 'T' where t>T.Will this object remain at temperature t forever if we don't disturb the system or will it emit energy by heat until an equilibrium is established between the object and its surroundings? If it does so then why does it do so?
 A: There are three processes that will modify the temperature of your $t$ body.


*

*Firstly, there is diffusion happening between the $t$ body and its surroundings because their temperature is different. The flux of energy is imposed by the second law of thermodynamics : it goes from greater temperature to lower temperature ($t$ gives energy to $T$) until both temperatures becomes equal.

*Secondly, the $t$ body emits energy in the form of photons : this process is called blackbody radiation and it would happen even if the $t$ body was surrounded by nothing but vacuum;

*If one or two of the bodies is a fluid, its change in density caused by temperature inhomogenities causes a flow that can carry heat : this phenomenon is known as convection and also participates in the dynamics of the change of temperature of your $t$ body.
To understand why there is blackbody radiation, take the example of a body at temperature $t$ inside of a box filled with nothing but vacuum. For thermodynamic equilibrium to exist, the body needs to be in equilibrium with the vacuum, and this vacuum can contain photons. The blackdody radiation comes from the equilibrium between the gas of photons inside of the box with the matter body at temperature $t$ : this body interacts with the electromagnetic field by absorbing and emitting photons according to the thermodynamic laws of blackbody radiation.
A: I assume that by emitting heat you mean infra red radiation?
A body emits radiation.  The higher the temperature the more radiation is emitted.  If you have a body ay a temperature t whilst the surroundings are at T with t>T then the body will radiate energy whilst also receiving less energy from the surroundings.  So the body will suffer a net loss of energy.  Infra red radiation originates from the vibrational and rotational motions of molecules.
A: Everything emits thermal radiation, proportional to fourth power of temperature, following the Stefan's law:
$$j=(1-a)\sigma T^4$$
where $a$ is albedo. For $a=0$ this is black body radiation. Of course, there are no perfectly white surfaces, so this can't be exactly zero -- still, reflective surfaces help to diminish radiation (the silver coating on your Thermos is essential, the vacuum itself wouldn't help that much if the walls were black).
Because of this, nothing can help you completely insulate the system thermally. Even through vacuum, heat is exchanged through radiation (that's what the Sun is doing, and above formula is how we know how hot its surface is). Also, fire, lightbulbs and thermal cameras, everything works off this principle.
The spectrum of this radiation of course depends on the temperature too (Planck's law), but just for computing the radiative power, above formula suffices. The fourth power ensures that when you're hot enough, the losses are enormous (that's why hot things glow).
A: There are two phenomena which let emit every materia electromagnetic radiation.
First, above the temperature 0 Kelvin the atoms of every materia are not in the lowest possible state and due to this excited state the atoms emit photons. This is its nature and don't have a deeper explanation.
Second, nothing could be isolated perfectly from the surrounding EM radiation and due to this it's not possible for materia to come to the lowest possible state. In any case the cooled down materia will be hit by photons.
No one body reach ever 0 Kelvin. No difference, if one place a cooled to say -100 °C body in a room of 20°C or if one place a cooled to say 0.001 K gas in an isolated volume of 0.01 K. In both cases the surrounding environment (the room, the volume) are emitting EM radiation and some of the photons hit the cooled body and increase its temperature.
This happen even if both constituents have the same temperature. In this case they emit more and more photons of lower energy (since perfect reflection does not exist). How long this process takes is unfortunately not observable because the two constituents are not an isolated system too and under the influence of the surrounding environment.
A: If you are considering an object placed in anything but a vacuum (let's say a glass with hot water in a room at normal temperature for example) 
This object temperature will decrease towards T until equilibrium. This comes from the empirical Fourier's law of thermal conduction, which states that any temperature gradient (= temperature difference across space) will cause a heat transfer :
$$
\vec{\phi} = -\lambda \vec{grad}(T)
$$
$\vec{\phi}$ being the heat flux and $\lambda$ being the thermal conductivity.
A: Object temperature correspond to gigling motion, which of course transmits mechanically to anything in contact: a first exchange - and thus trend to balance - occurs there.
Then there is also an internal balance between the various degrees of freedom, one resulting in emission of IR, in any body (i.e. object as surrounding). if unbalanced, one will receive IR energy, which will contaminates it's mechanical energy, i.e. increasing heat, up to balance.
