She never turned up. What options do I have?
Since you have already agreed to sign the joint options, what else is the option you expect?, you have exhausted your option by agreeing to sign the joint petition, so you may have to follow the same route or decide to agitate her decision.
1) She has home key. Can I prevent her from entering my house ?
Until she remains your wife, you cannot prevent her from entering your house.
2) Can I prevent her cousins from entering my house ?
Yes, you can very well prevent them from entering your house, if they still do that, you can give a criminal complaint against them for criminal trespass.
3) Can I put a legal pressure on her to take care of the kids?
They are your kids too, you have an equal responsibility to rare the children, so you cannot put pressure o her for this.
4) Can I apply for divorce alone? I mean without joint petition.
Even she is giving you the same option, i.e., divorce, what is that you are going to achieve by filing a separate divorce petition on your own?
If you have decided to divorce her, it will be better to go for mutual consent divorce especially when she is voluntarily agreeing for it.