Suppose all your covariates are continuous, each "Coef" represents the log odds ratio when each covariate is increased by 1 unit, while holding other covariates constant. "EXP(Coef)" is therefore the ordinary odds ratio.
To answer your first question, ideally you should not consider "Coef" or "EXP(Coef)" only, the standard error must be also taken into account. But if we assume the parameters can be estimated equally accurately, then for "Coef", you should compare the magnitude, namely, absolute value. For "EXP(Coef)", you should compare its "distance" from 1 (odds ratio = 1 means independence). The larger, or smaller the number to 1, the stronger the correlation is.
For the second question, "EXP(Coef)", i.e., (estimated) odds ratio, is of course always positive regardless the "Coef" is positive or negative. However, if "Coef" is negative, then "EXP(Coef)" is between $(0, 1)$, representing the response and covariate is partially negatively correlated. As above, when viewing it, you should compare it with 1 (instead of 0) to assess the covariate's effect. For example, 0.0159 should be considered a very strong association, as $1/0.0159 = 62.89$ is way bigger than 1.