Pay attention closely to breaker specifications
As it turns out, you don't need a fuse in the solar panel wiring provided you use adequately sized wire from the solar panel to the charge controller. This is because solar panels are inherently current limited sources; in your case, your panel specifies an Isc of just under 6A, so it won't put more than that out, no matter what happens, meaning that 1.5mm2 wire is plenty adequate for the panel connections.
However, the battery wires are a different story. A fully-charged lead acid battery can put out somewhere between 1000 and 5000A of short-circuit current, and you will need breakers or fuses that can handle such a fault at 12VDC. This means that you will need to check the specifications for the breakers you want to use; not only will you need a 50A and a 100A breaker, these breakers will need to have a minimum of a 5kA interrupting rating at >16VDC. Some IEC-style breakers are DC rated, but not all, so you must check the specifications for the breakers you want to use.
As to wire sizing, you'll need some fairly thick wires for the charge controller and the inverter. 10mm2 is a reasonable size for the wiring from the charge controller to the battery; however, you'll want to increase that all the way to 35mm2 for the wiring from the battery to the inverter as the inverter can pull much more current than the charge controller provides. You also may need to use fine stranded wire for the connections to the battery to avoid putting mechanical strain on the battery posts, which requires its own precautions for making good connections (such as using compression lugs instead of mechanical setscrews). You'll also want to connect the negative post of your battery to an earth ground stake, or use two-pole breakers if you want to leave it floating.
Don't get gassed!
A vented-type lead-acid battery, like yours likely is, releases highly flammable hydrogen gas during the charging process. This means that you will need to make sure that the battery lives in a quite well-vented space so that hydrogen can't accumulate and ignite.