The Earth's magnetic field is generated in a different way from the magnetic field in the permanent magnets we use on the surface. It is generated by flowing currents of molten iron in the outer shell of the core. This does not create a very large magnetic field (around 25 gauss at the surface of the core), but because the core is so massive, the weak field is very large in space. Out here, at the surface, the field strength is only 0.25 gauss to 0.65 gauss, because we are so far away from the core where the magnetism originates. A strong permanent magnet like N45-grade supermagnets are made out of a highly magnetic material called neodymium iron boride. This materials is way more magnetic than the iron in the Earth's core, and a small supermagnet can easily generate fields up to thousands of Gauss right next to them. However, because the magnets are not very big (compared to the Earth's core), that field is not very big spatially, and decays within a few inches or feet. So, Earth's field is a big, weak magnet and N45 magnets are strong, small magnets.

How strong is the magnetic force that holds atoms together 00.03645 Gauss?

Atoms aren't really held together by magnetic forces. Rather, the nuclei are held together by the strong nuclear force, which is much stronger than a magnetic force. The electrons are held to the nucleus by electrostatic interactions, which are also much stronger than the magnetic forces inside of atoms.

Also I would like to try to build a permanent magnet motor to power a generator, is it possible?

I'm not sure I totally understand this question. A generator and a permanent magnet motor are basically the same thing. If you take a motor and attach its leads to a voltmeter and then spin its shaft, you will see a voltage. This is using a magnet as a generator. You can build a motor or a generator using a permanent magnet and some magnet wire (which you will have to wind into coils). However, if you try and use a motor to power a generator. You will have to supply electricity to make the motor turn, which in turn will turn the shaft of the generator to make electricity. But, since the motor and the generator are both less than 100% efficient, you will always get less energy out of the generator than you supplied to the motor.

Earth's magnetic field is weak because the electromagnetic dynamo that generates the Earth's magnetic field doesn't actually propagate that much electrical current, which means that the magnetic field created by said current is weaker than with a magnet created specifically to be a magnet.

Magnetism does not hold atoms together. The electrons in an atom are held together by electromagnetic force, but the force consists of quantum energy states referred to as 'orbitals'. A strong enough electrical field will strip the electrons off of atoms (this is what causes lightning), but I am not sure how the dielectric constants of matter that determines the voltage per unit distance required to create an electric spark works at the quantum level.

I am not sure what the number 0.03645 Gauss refers to, or what a "permanent magnet motor" means. If you are trying to build a perpetual motion machine (a machine that produces energy without consuming some kind of potential energy), then that is impossible, as it would violate both the first and second laws of thermodynamics.