you have got the general concept right, except crossovers involve 2(or more) genes at separate loci. One well known poultry example for crossover is P for pea comb and O for the blue eggshell gene. As it happens, those two genes are located on loci very close to each other so they rarely 'separate' because genes tend to break off in sections instead of every single gene being 'separated' and recombined completely. Think of chromosomes being split up in ribbons of different sizes and you get the general concept. The P and O are so close they almost always are on the very same 'cut off ribbon section' during recombination. Result is almost all pea combed hens with this linkage turn out to be colored egg layers, even if crossed/mixed with single combed birds. Just about 99% of the time. It works so well if you are planning to only keep colored egg layers you can cull the single combed chicks at hatch.
Occasionally, two 'linked' genes happen to be 'cut off' in between during recombination. That's called crossing over- one of the previously linked gene gets 'separated' and is 'transferred' to another chromosome. In the case of colored egg(blue or green), pea combed birds mixed with non-colored egg(white or brown), single combed birds, a single combed pullet that lays colored eggs would be evidence of a crossover. In her case, P and O happened to be cut off in between and the O gene 'moved'(crossed over) to the chromosome with p+. making her p+/p+ O/o+.
It can happen the other way too, P and O split up but P crosses over to chromosome with o+ resulting in pea combed laying white or tan egg. This one is harder to be sure of as there are plenty of P o+ birds so it could have been a P o+ chromosome bred into P O linked flock. Single combed colored egg layers are much less common so they stick out like a sore thumb and much easier to be sure of a cross over.
p.s. to answer your I example, yes the same loci can produce completely new mutation, or a mutation can mutate again. IIRC, Smoky is a mutation of I.