
Students should be very careful these days and really take the time to read the Terms of Service of the company they are considering hiring. There are actually more and more companies that opt to retain their rights (the writers give up these rights to the work as soon as the paper is submitted) to the paper for future use. I noticed a rise in this copyright retention with the companies when an upsurge in "free essays" within these companies began. Apparently, they keep the rights to "revise, alter and republish" the paper in any manner they deem fit within a given time frame. That means, they can republish the same paper, with slight changes to the content, a semester after it was originally ordered by the student.
Unfortunately, there is no guarantee that a company that declares a transfer of ownership of the paper to the student will actually honor their word. Writing companies have been known to pay their writers to paraphrase the same paper for them. They pay the original writer to change the wording and as such, the writer gives the company a totally new paper, that can be sold to a new student based on the same assignment instruction.
The company will not use the same paper within the same semester. The company will also not use the paper in the same form. The original writer will be ordered to revise the whole document either as a revision request or a small change paraphrasing order. You don't have to worry about the company using your paper immediately after you bought it. They will give you at least one semester to use the paper free and clear of any plagiarism worries.
As for the second question, most writers that I know of automatically turn over the rights to the complete paper to the student. As @Smiley73 said, the paper was originally written for you. The writer doesn't have any interest in retaining your order simply because he will have to write a totally new one for a new client, regardless of how similar the order instructions may be. Due to the length of time that has passed, the writer will at least have to update the sources, in which case he ends up writing a new paper anyway. So no worries when it comes to the copyright of academic papers. It always rests in the student's hands.