I need an advice: should I got for a nice short domain with no keywords (but somehow related to the site content) or it's better to register a long keyword rich domain name? What's your opinion?
If short domain is brandable go for it. It will not be good for short term, but in long term it will help.
When you mean long, how long is it ? 2 words, three words ? And how competitive is the niche which contains these keywords ??
I use several 2 words domain and I perceive effect of this to be positive on SEO. I can not judge what is the effect on customer behavior.
At least one main keyword helps in the short term as noted by jimkarter, but if you're planning to promote the hell out of a site and its catchy go for the branding term URL. Think about it like this YouTube Amazon.com ZDNet Oh, and let's not forget, Google
Short but catchy brandname is advisable for long term gains. Looking for long term gains will be more profitable as you have to put in quite good amount work in it.
I think you should have your keywords in your domain name, in short, keywords in a domain name is important because it is still recognized by search engines.
You should also think about user behavior. When users see a domain name like www.TheCheapestCarInsurance.com, they automatically are keyed in on your topic, especially if they are searching for inexpensive automobile insurance. But, if you try for a brandable name, something like www.PizoogleTube.com, you will have to spend a lot of money and time associating PizoogleTube with inexpensive auto insurance.