Refused entry to India

The answers to your questions can be found in Air India's Conditions of Carriage, which you agreed to when you purchased the air ticket. Specifically:

1. Can I blame the airline that let me board without checking my visa at the point of origin (Tel Aviv). Can I claim also compensation due to the 32 hours trouble?

No. Article 14.1 states that it is your responsibility, not the airline or the travel agent's, to ensure you are properly documented, including making sure the relevant visas are linked to the correct passport. This responsibility remains with you even if you permit another person to apply for such documentation on your behalf.

2. According to IATA Air India needs to send me back on their expenses.

This is incorrect. Article 14.3 states that you are responsible for paying for the expenses of your removal. If you have purchased a return ticket, this is typically done by using the return portion of your ticket to send you back.

It is possible that Air India may have been fined by the Indian Government for not checking your visa prior to boarding. Under Article 14.4, Air India has the right to recover these fines from you.

3. I need to go back to India and I have already a new e-visa in place.Can I demand that Air India will issue the new ticket for free?

You can always demand. However, given the two points above, it is very unlikely that Air India will accommodate them.

It should be noted that these types of conditions are typical for all airlines, not just Air India.

You are looking at a contract that is none of your business. Yes, the IATA rules are binding between the airline and the airport. If a passenger is refused entry at the airport, the airline, not the airport, is responsible for deportation. However, you are not a party of this contract.

The airline has a binding agreement with the person that booked the flight (depending on jurisdiction and contract, either you or the travel agent), called conditions of carriage. During booking, one has to accept these conditions. They allow the airline to recover costs they incurred in such an event. Since you booked through a travel agent, either your travel agent did book without notifying you about the conditions of carriage, or they did, but you didn't read them (maybe even signed without reading). Also, all travel agents I know have own terms of service, which should tell you how much liability they take. Furthermore, depending on jurisdiction, the terms of service or conditions of carriage may be fully or partly overruled by local consumer protection laws.

Your recommended route is to talk constructively to your travel agent about their mistake and how they may fix it; if that fails, to search for jurisdiction-specific landmark decisions, and then lawyer up and go to small claims court if you're willing to take the risk.