I know that links which come from relevancy site can help your site ranking. If your site is about dog then you should gather some links from dog related sites. But my question is, Is it true or just our assumption? Is it possible for link which come from old site but off-topic can help your ranking as same as link cames from on-topic site?
All links are weighted. Most of them will carry some weight but all other things being equal (PR, etc.) links from relevant sites and relevant pages will carry more weight and thus help your SE rankings more. You might want to take a look at some free or reciprocal web directories in your niche to give your site a little boost. Here's a list of niche directories to get you started.
you will get credit for the links more or less. its just on on topic site, its also the site's overall ranking that is passing you the link. idea is to get as much links as you can get
you have to be carefully analyze the site that you're going to have a link exchange. because sometimes by doing this with inadequate knowledge that site could drag your site down, instead of pull your site up!
Relevant sites pass more SEO weight to your site, relevant backlinks have nothing to do with Pagerank, so consider Pagerank out of the relevant backlink theory. The reason relevant links carry more weight is because the on page content where the backlink resides is related to the anchor text and the link it's pointing to, which would mean the link has more value then if it wasn't related. Matt Cutts has stated on his blog that relevant links carry more weight.