Crossing over is not rare, though it varies from species to species. In humans, it occurs on average once per chromosome arm per meiotic division. In other words, during one meiosis, you'll get around 46 crossovers in a human stem cell. But nonrecombinant chromosome pairs - those that don't undergo recombination - are not rare. They happen all the time. In fact, my PhD project is studying exactly how that works.
To your second question, it is technically true, but very uncommon. Far more common is for a pair of chromosomes to start undergoing recombination, and then back out and go back to being separate. Also, depending on how the Holliday junction is resolved, only a tiny bit of DNA may actually be transferred from one chromosome to another.
Recombination frequency is a measure of how often you see two pieces of DNA go to different spindle poles. If two pieces are completely independent, they assort randomly. Because there are two poles, half the time they go together and half the time they separate. Thus, 50% recombination frequency. If you have two pieces of DNA on the same chromosome that have a 50% recombination rate, that means that they're so far apart from each other that basically just about every time a cell goes through meiosis, there will be recombination between them.