When I do a preview on premium themes, I don't see pictures on the posts I have made in the past. Why is that?
Which themes were you previewing, specifically? Where were the images not showing up -- e.g. on the front page or on single posts? I ask because themes vary in the way that they display images. For example, some themes display thumbnail images on the front page, and some do not.
And if I were to purchase a premium theme, would I have to adjust all of my old posts so they show up properly?
No, you don't need to adjust the old posts. The theme will display your posts automatically. Some themes contain options pages that let you configure how posts display on the front page, but you won't need to edit the posts themselves.