Dear Querist
My opinion on your queries are as under:
1. What does the first sign and 30 day period and then 2nd sign in court marriage mean?
Opinion: when the parties file an application for registered marriage then there is a rule for affixed their application before marriage registrar office and wait till 30 days for any objections, if there is no objection then only the marriage will be solemnized, at the time of marriage the second signed of both the parties are mandatory with three witnesses along with the sub-registrar/marriage registrar
2. Does marrying in court but not registering still considered one as married?
Opinion: apart from the above, without following above procedure the marriage can not be valid.
3. I want to know if her marriage before ours is still valid & does that make my marriage with her void?
Opinion: yes, if she is already married then your marriage with her is void ab-initio
4. On what grounds can I file divorce? Or I can file for annulment of marriage?
Opinion: If she is already married then file annulment petition otherwise i don't think so that there is any reason for divorce as per the ground mentioned in the act.
5. Can I file a fraud case against her & her parents?
Opinion: if she is married then you may file a cheating/fraud cases against them before criminal court under section 420 of IPC.
does that mean it was marriage even if it was not registered?
Opinion: Non-Registration of marriage is not means that the marriage is not registered, the marriage will be valid.
There must be some details in marriage registration office about this event took place, Is there any way I can acquire the proof of registration from Marriage Registration office?
Opinion: Yes, there is a provision, you may file an RTI before Sub-Registrar office for got the information with her detail, if there is any information is avaliable then they will provide the same to you within 30 days.
If at all succeed to acquire the proof can I file divorce on the basis of it?
Opinion: If you get the proof of her marriage registered with the registrar or otherwise, then file an annulment case u/s 12 of Hindu Marriage Act, 1955
you may also file a criminal complaint against her u/s 494 of IPC for Bigamy.