How can that temperature kept for 4 billions of year?
The Earth is big, and rock isn't a very good conductor of heat.Does the Earth core produces its own energy?
Yes, it does. The Earth's inner core is growing. To melt something, you need to supply heat. Melting is an endothermic process. The opposite, freezing, is an exothermic process. The growth of the inner core releases latent heat. Gravity also plays a role. This freezing selectively concentrates iron and nickel in the inner core, leaving lighter impurities in the molten outer core, thereby releasing gravitational energy.
Most (but not all) geophysicists are of the opinion that there is very little radiogenic heating in the Earth's core. The four isotopes responsible for almost all of the Earth's radiogenic heating are uranium 238, uranium 235, thorium 232, and potassium 40. Chemically, uranium, thorium, and potassium are lithophile elements, which means they are found in rock (not in the core). Moreover, they are incompatible elements, which means they are concentrated in the Earth's crust. Those elements should be very strongly depleted in the Earth's core. (Some geophysicists think there may be some potassium 40 in the Earth's outer core.)Can the temperature heat the mantle and Earth surface through convection heat transfer?
The Earth has a magnetic field. Two things are needed for this: A rotating fluid that conducts electricity, and convection in the fluid (i.e., a heat flux through the fluid). The Earth's core has to be heating the mantle; if it wasn't, we wouldn't have a magnetic field. Mars has a liquid inner core but doesn't have a magnetic field; this is presumably because Mars has a stagnant lid rather than plate tectonics.
The amount of heat flux needed to sustain the magnetic field is subject to debate. Different geophysicists agree to disagree. Most think it is about 5 terawatts across the core-mantle boundary. Others think less is needed. How much heat actually flows across the core-mantle boundary is also subject to debate. Most geophysicists think it is between 5 and 15 terawatts. Some argue for a smaller number, others for an even bigger number, making the range 2 to 17 terawatts. As an interested bystander, those widely varying numbers means that a lot more work needs to be done.