A copy of the FIR would help to figure out what all allegations under what all sections have been included in the FIR. Merely filing a complaint with the police does not prove your offence and the police do not have the authority to summon you only the basis of just a complaint. Had you been here in India, the police could have arrested you under 498a. But since you are staying abroad, this is an intimation for you to take necessary counter legal measures. Moreover going by what you have mentioned that your wife left your household 1.5 years ago, and that your children are happily and willfully staying with you, it would be difficult for your wife to press charges of kidnapping against you.
As you have also mentioned about a mutual agreement, if you have any document related to the same, it would help you in all legal proceedings against you.
To specifically answer your questions :
1. In case LOC has been issued, then a bail is required for you to be able to travel back to India, else you could be arrested / detained.
2. You can reply to the email that you have received from the police station in India and seek details and copy of the complaint / FIR, as this is your legal right. Moreover this would also help in gauging whether the email is genuine and whether it has come from the authorized source i.e. the concerned police station as claimed in that email.
There is no official centralized mechanism for finding out whether an LOC has been issued against an individual anywhere in India. Any local advocate in the city of the police station where that complaint has been filed would be able to help you out in coordinating with that particular police station to find out whether any LOC has been issued.
3. In case an LOC has been issued, then you would have to apply for and seek bail before you land in India. For this, you would have to send a Special Power of Attorney to your advocate here in India, who would then be able to assist you in procuring the bail.
Generally LOC is not issued for such trivial matters. It is issued for grave crimes and cognizable offences, or in cases where the accused is either not traceable, refusing to receive and accept summons, or is purposefully trying to avoid and evade the law.
If you are sure and get a copy of the FIR, then simultaneously you would need to apply for quashing of the FIR in the High Court of the state of the concerned police station. Once the FIR gets quashed, then all legal proceedings related to that complaint / FIR including the LOC (if any) would also get terminated.
4. As already stated above, a local advocate in the city of the police station where that complaint has been filed would be able to help you out with any enquiries related to this complaint. The cost varies and would depend on the charges that your advocate would seek.