Does linking to websites that are relevant to the subject of your website without receiving a backlink hurt your SEO, help it out, or doesn't really do anything?
It should actually help ... for example, if you write a page about widgets, and link to wikipedia's page about widgets, it's actually a good thing for you (of course not as good as the other way around - if wikipedia's page about widgets linked to yours). You may want to consider putting the rel=nofollow attribute into those links, though.
Only if you have so many links that you seem like a link farm, which generally won't happen unintentionally.
I made an experiment once, registered a domain name, put a page with a description about what google is and linked to homepage and to some internal pages of theirs. That page got PR3 next update. I didn't promote it at all. I wonder if that PR comes from the outbound links I had on the site
I think it helps to improving your PR. But relevant sites should have good stats and they must not be Google's enemies!
still, IBLs to OBL had a difference in SEO. Much prepared to have IBLs than OBLs unless, if you have a link back in return.
Having outbound links to relevant-high quality sites is definetly a ranking factor! So don´t worry about your outgoing links - although i would recommend not to link to too many sites, like moneymoose said: don´t look like a linkfarm!
it wouldnt hurt as long as it is pointed to a quality webpage but of course it is much better if your outbound link is pointed in a related niche because the anchor text of that link can also help you with your targeted keywords.