Hi, neighbor!
The cable size to the bus bar depends on the current flowing through the cable and the length of the cable. Figure those out and choose a cable size appropriate to the situation. The main purpose of fuses is to protect wires etc. from overheating and causing a fire in the event of a short circuit or overload. With that in mind, the connection from the battery to the bus bar should be protected, as it's not beyond the realm of possibility for the bus bar to get shorted to ground (it's comparatively exposed). Every connection to the bus bar should typically also have a fuse or other overcurrent protection because generally the wire for that connection wouldn't be heavy enough to carry the full current that could flow through the battery and bus bar connection. (This is, of course, only pertinent to the positive side. Shorting the negative to ground isn't a concern.) I would also suggest figuring some way of enclosing the bus bar to prevent accidental shorts. This doesn't need to be anything terribly fancy or exotic; a sturdy plastic or wooden box would be fine in my opinion.
There's no inherent reason why you couldn't connect the converter or an inverter to the bus bar system rather than directly to the battery assuming everything is properly sized to minimize voltage drop to an acceptable level. There are some definite advantages to having only one connection to the battery; it's easier to avoid messing up when making connections, for one.
The negative side does need to connect to the chassis ground somehow. If you use a negative bus bar, that could reasonably be two heavy-duty connections to it (one to the battery, one to ground). There are other acceptable variations.