1. For a cross sectional study it is not necessary do a logistic regression

2. Even if we do we say the odds ratio as crude odds ratio

3. Hazard ratio is the Relative Risk averaged over time.

Yes, it is good to use Cox Model for analysis of cross-sectional study.The following BMC methodology paper nicely compares the different approaches to analyse cross-sectional data.

http://www.biomedcentral.com/content/pdf/1471-2288-3-21.pdf

If you are assuming that the event has constant chance of occurring within a time period over the time frame your study, then Cox PH regression will be fine. The basic assumption of Cox PH regression is that the hazards remain proportional over time. 

Next thing is you may want to investigate the reasons for large confidence interval. Here are possible reasons:-

1) Rare Event

When the number of events in one group is low (or even no event) then you'll have a large estimate of variance and hence large confidence interval.

2) Model Convergence

Has the numerical method used to estimate your maximum likelihood estimator converged? When convergence hasn't been reached and the fitting algorithm interrupted, the particular estimate which might be used to estimate the variance may not be optimal.

HTH

Yes, you can do it in cross-sectional study, when your odds ratios are too high in social science. But, you must be cautious in your interpretation.

Hi,

Hazard Ratio model:

In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two levels of an explanatory variable.

For example:

In a drug study, the treated population may die at twice the rate per unit time as the control population. The hazard ratio would be 2, indicating higher hazard of death from the treatment. Or in another study, men receiving the same treatment may suffer a certain complication ten times more frequently per unit time than women, giving a hazard ratio of 10.

Hazard ratios differ from relative risks in that the latter are cumulative over an entire study, using a defined endpoint, while the former represent instantaneous risk over the study time period, or some subset thereof. Hazard ratios suffer somewhat less from selection bias with respect to the endpoints chosen and can indicate risks that happen before the endpoint.

Thanks Philippe.

Correct, We wont be able to fit any cox regression model at all. Yet, one can fit a logisitc model which will result in a large estimate of standard error.

Here is an example of what I mean using R.

set.seed(92828) # for reproducibility

# fake data

test table(test$group,test$resp)

   0 1

A 10 0

B 7 3

And  now if you fit a logistic regression, you see what the results below:

model1|z|)

(Intercept) -19.57 3400.72 -0.006 0.995

factor(group)B 18.72 3400.72 0.006 0.996

You see the standard error of 3400.72, which is manifestation of the fact that there were no events in Group A. 

HTH