It's good that you placed the demographic questions at the end of your survey. This is an example of good survey design and demographic questions -- especially the household income question -- tend to be more sensitive and survey participants are frequently less likely to complete them than less invasive questions. This is why they are usually placed at the end.
To answer your questions, yes, you can use only the partially completed data, but if you do so, you should be clear in your presentation what how many people answered the questions you are reporting on (i.e. what the denominator is) as this value changes as participants in your survey dropped off. You should provide an overall completion rate (how many answered all question) and then for sub-analyses, indicate how many completed that question or questions for bivariate analyses. The American Association for Public Opinion Research has more detailed explanations of how to report response rates and provides a calculator to help you compute them: http://www.aapor.org/AAPORKentico/Education-Resources/For-Researchers/Poll-Survey-FAQ/Response-Rates-An-Overview.aspx
Most people will be concerned about the generalizability of your findings if a lot of people refused to complete the substantive portions of your survey. The demographic questions can often be used as a gauge to the generalizability of your findings (e.g. if you have some people completing the demographics but not certain questions, you can use the demographic data to compare to those who completed the survey in it's entirety to make a judgment call about whether or not those who completed all questions are in some way different from those who only partially completed questions).
If you can obtain data on the characteristics (perhaps in your sampling frame) about those who abandoned your survey, it would be recommended to compare those characteristics to those who completed. If there differences are minimal, you would have increased confidence in the generaliability of your findings.