If I have alleles AB on one chromosome and ab on another, and if A and B are far from each other (and also a and b), then there is a lot of chromosomal crossovers happening. If I crossover 7 times, will I end up with the same chromosomes as if I cross over once?
Good thinking. Yes, it is theoretically possible. It is often modelled like this. Have a look at this answer for the maths.
Note however, that to my understanding (and I might be wrong) there is typically either 0 or 1 crossover per pair of chromosomes as cross-over is directly related to the way homologous chromosomes bind together during prophase I of the meiosis by the synapsis. My confusion lies in the fact that, I think, there are sometimes (often?) several synapsis per chromosome pair. Hopefully, someone else will be able to clarify this for us.
As a consequence, the genome-wide recombination rate in humans is almost exactly 23 in humans (Wang et al., 2012). So in practice, such mulitple crossover would very rarely (if ever) happen.
That's quite an interesting question. Crossing over occur during meiosis, there can be more than one in a chromosome indeed. Let's say two crossing over occurs in the same chromosome (the one with your markers A and B), if they both occured in between your markers, you will see no difference. As you said your markers are far from each other you have a good probability to observe that. If your markers were close, your chance to observe a double crossing over is really low.
Also, there is some mechanisms preventing two crossing over to occur too close from each other. I can't tell you exactly what it is about. There is a type of Crossing over not subject to this rule though...
Double crossing over can be an issue when working on genetic maps (using crossing over to determine the order of marker on map). As if this occur we have no way to choose between 0 crossing over or 2 crossing over (meaning markers are actually far from each other.
If you want more explanation about this wide topic, I would suggest you to read this scientific work. They are working on removing inhibition for crossing over and achieve to multiply the number of crossing over by 9 in Arabiodpsis!
Synapsis is the pairing of homologous chromosomes, one from each parent. Crossover occurs following synapsis. The points at which crossover occurs are called chiasmata, and appear to be genetically determined. However, it seems that they are not always at the same locations on homologous chromosomes from different individuals.
That said, the likelihood of crossover being reversed on one chromosome would depend on a series of lucky coincidences and some serious inbreeding, but could conceivably happen. For crossover to be reversed on all chromosomes would be staggeringly unlikely.