Do transposons usually jump from one chromosome to another?
Yes of course; they are not like tandem duplications. 

However, some transposons have site preferences; for example the Tn7 transposon.



EDIT:

I don't know what you mean by original location. Some transposons, especially LINEs, have expanded a lot in the human genome. For any individual you can check the germ cells for a reliable estimate on the original positions. There are mechanisms such as piRNA mediated silencing that suppress transposon activity in germ cells. The genome of the germ cells also tells you what exactly will be inherited by the offsprings.