In this question and in two other recent questions ([here](http://stats.stackexchange.com/q/228229/28500) and [here](http://stats.stackexchange.com/q/227068/28500)) you are interested in the "slope" that represents the relation of a continuous variable, `NE`, to survival in a Cox proportional hazards model. You are particularly interested in whether the relation of `NE` to survival differs between sexes; furthermore, individuals might belong to either of 2 Groups, `G1` and `G2`. I'll assume that linearity with respect to `NE`and the proportional hazards assumption are both verified. I'll ignore the random-effects term* that you included in other related questions.

Much depends on how you want to treat the group membership issue. In the example of this question,** where there is little evidence for differences in outcome with respect to Group `G1` versus `G2`, preliminary data exploration and subject-matter knowledge might suggest that you simply remove the `Group` variable from your model. Tests of the interaction term `SexM:NE` would then provide the answer to your question about sex differences with respect to the relation of `NE` to survival.

If you wish to maintain the breakdown by `Group`, you might consider using the `anova` wrapper for the Cox model output in R to provide a single test of the `SexM:NE` coefficient, combining information from both Groups. The `anova` function performs a hierarchical test of coefficients in the order of entry into the model, using the equivalent of [Type I sums of squares](http://stats.stackexchange.com/q/20452/28500). If the data are reasonably balanced among combinations of covariates (for survival models, in terms of event numbers) then this may provide a useful test provided that you are clear about what it examines. For example, if you specified the following model:

    Surv(time,status)~ NE + Sex + Group + NE:Sex + NE:Group + Sex:Group + NE:Sex:Group

then `anova` would first associate as much as possible with `NE`, then with `Sex`, then with `Group`, and then test whether the `NE:Sex` interaction significantly explained any residual. Note that this is a different way of evaluating the results of the model from the treatment-contrasts summaries provided by `print` or `summary` that you display above, even though it is based on the same model.

If you have a particular interest in the 3-way interaction among `NE`, `Sex` and `Group` then you will need to examine contrasts of the coefficients, similar to what you propose but with an important difference in implementation. The test you propose is equivalent to a [Wald test](https://en.wikipedia.org/wiki/Wald_test). Examining whether there was a difference between males and females with respect to the value of the `NE` coefficient in Group G2, you would test whether $\beta_{SexM:NE}+\beta_{SexM:GroupG2:NE}$ (`MG2-FG2` in your question) is different from 0. As the coefficient estimates are correlated, you need the [variance of a sum of correlated variables](https://en.wikipedia.org/wiki/Variance#Sum_of_correlated_variables), which in this example is:
$$ \mathop{\rm var}(\beta_{SexM:NE})+\mathop{\rm var}(\beta_{SexM:GroupG2:NE}) + 2\mathop{\rm cov}(\beta_{SexM:NE},\beta_{SexM:GroupG2:NE})$$

Your proposed Z-test (based on the square root of the variance) ignores the covariance of the estimates of the coefficients, which is necessary and is provided by the corresponding off-diagonal element of the variance-covariance matrix, which you can get from the `vcov` function applied to your model. If you use the `rms` package, then there is a `contrast` function for that package's `cph` Cox models that allows tests of arbitrary contrasts, including bootstrap non-parametric tests.

One warning on this approach, however: Terry Therneau, responsible for much of the survival analysis infrastructure in R, has warned in [this vignette](https://cran.r-project.org/web/packages/survival/vignettes/tests.pdf) that the apparent similarity of Cox regression models to linear regression models does not necessarily extend to tests on contrasts of coefficients. Examine those arguments carefully as you proceed.

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*In another question you note that you are also evaluating a "random effect" that only has 2 levels. Treating a variable with so few levels as a random effect can be considered inadvisable. 

**The model results in this question differ from those in other questions you have asked recently on the same general matter. It helps to specify the seed for the random-number generator, e.g., with `set.seed()` in R, before you generate your random data, to have reproducible "random" data for demonstration.