I'm going to answer your questions out of order:
3 Would deleting my nonevent population would be good for the accuracy of my model ?
Each observation will provide some additional information about the parameter (through the likelihood function). Therefore there is no point in deleting data, as you would just be losing information.
1 Does accuracy of logistic regression depend on event rate or is there any minimum event rate which is recommended ?
Technically, yes: a rare observation is much more informative (that is, the likelihood function will be steeper). If your event ratio was 50:50, then you would get much tighter confidence bands (or credible intervals if you're being Bayesian) for the same amount of data. However you don't get to choose your event rate (unless you're doing a case-control study), so you'll have to make do with what you have.
2 Is there any special technique for low event rate data ?
The biggest problem that might arise is perfect separation: this happens when some combination of variables gives all non-events (or all events): in this case, the maximum likelihood parameter estimates (and their standard errors), will approach infinity (although usually the algorithm will stop beforehand). There are two possible solutions:
a) removing predictors from the model: though this will make your algorithm converge, you will be removing the variable with the most explanatory power, so this only makes sense if your model was overfitting to begin with (such as fitting too many complicated interactions).
b) use some sort of penalisation, such as a prior distribution, which will shrink the estimates back to more reasonable values.
Best Answer
Re 1: If you predict well in the hold out sample then you're doing well (no time to worry about propriety ;-) But since you're asking...
One way to look at the threshold is that when you set it to 0.1 you are implicitly specifying a loss function. That is, separating the question of what to do (e.g. approach a customer) from what to infer (e.g. that the probability is of 1 is 0.15). Indeed, you might make this separation a bit more explicit in your question. For example, you talk about needing to approach 5% of some people for something to be worthwhile. And then about how well you can predict cases. Is the issue that to approach the `right' 5% (presumably the true '1's) you might have to approach many more (true '0's) to no effect? Then the cost of approach is relevant and the threshold should be set to minimise loss. But you also say you can predict the held out cases well when the threshold is set at 0.1...
Re 2: The cause of low probabilities is an unbalanced category distribution. This may cause estimation problems, though don't automatically assume that it will. If it does you can often correct them quite easily by changing the training data set structure and correcting parameters or in other ways. There's some discussion here, a link to a good paper, and much more discussion elsewhere in the site - just search for 'unbalanced sample'.