I am not sure you have enough data to run logistic regression (normally you want a hundred cases plus more for each IV). But, if power is not a major issues, then logistic regression would certainly appear appropriate. It will give you something chi square won't, the specific impact the IV has on the DV.
I don't see how you can run linear regression with this data. While categorical data sometimes is used as a dependent variable (for example Likert scale data) the absolute minimum number of categories is probably 5 (many suggest ten seperate categories or more). It does not seem like you have this.
You could include the results of linear and logistic regression. But people would probably wonder why you did. Normally if you can run linear regression you do (it is much easier to interpret and has well understood diagnostics which are lacking in many cases for logistic regression or at least are harder to find in commerical software. For example a meaningful Rsquared). If your data does not support it, and I doubt yours does, then you should not do so.
People vary on what they think are important. Probably the most useful thing to report in logistic regression is the Odds ratio and the Wald test of paramater signficance. Also your overall model signficance test (I think that is a -negative two log liklihood test, but my memory fails me at times). Whatever commerical software you use should tell you where to find this (and what they call them). Some like the pseudo Rsquare tests as well (although not all by any means). Slopes are basically useless in logistic regression other than the sign and whether they are signficant (which is why Odds ratios and relative risk are used instead).
I suggest looking at "Using Multivariate Statistics" 5th ed by Barbara Tabachnick and Lidna Fidel. They have a chapter on logistic regression including data clean up. They are easy to understand and a wealth of information including on software for methods. Chapter 10 is on logistic regression.