Through trains between Hankyu and Sakaisuji

To travel from the Kyoto-Kawaramachi Station of the Hankyu-Kyoto Line in Kyoto, to the Nippombashi Station of the Sakaisuji Line in Osaka, Google map suggested the following:
- Take the Limited Express Osaka-Umeda train at the Kyoto-Kawaramachi Station of the Hankyu-Kyoto Line, to Awaji Station.
- At the Awaji Station, walk 2 mins to transfer to the Local Tengachaya train of the Hankyu-Senri Line.
- Continue on the same vehicle, as a through train, it will run on the Sakaisuji Line.
- Exit at the Nippombashi station of the Sakaisuji Line.

My questions are:
- If I use an IC card to enter the gate at Kyoto-Kawaramachi Station of the Hankyu Line, and exit the gate of the Nippombashi station of the Sakaisuji Line, will the fare for the whole journey (consisting of two railway operators/lines) be computed automatically?
- I do not need two tickets, right?
- Using my IC card this way crossing two railway lines will not brick it, right?
- At the Awaji Station, I am transferring from the Hankyu-Kyoto Line to the Hankyu-Senri Line only, I do not need to exit the gate and pay for the Hankyu-Kyoto section, and buy ticket for the Sakaisuji section and enter the gate again, am I correct?

Thanks in advance for your help.

- If I use an IC card to enter the gate at Kyoto-Kawaramachi Station of the Hankyu Line, and exit the gate of the Nippombashi station of the Sakaisuji Line, will the fare for the whole journey (consisting of two railway operators/lines) be computed automatically?

Yes, it will.

- I do not need two tickets, right?

No, you don't.

Using my IC card this way crossing two railway lines will not brick it, right?

No, it won't.

At the Awaji Station, I am transferring from the Hankyu-Kyoto Line to the Hankyu-Senri Line only, I do not need to exit the gate and pay for the Hankyu-Kyoto section, and buy ticket for the Sakaisuji section and enter the gate again, am I correct?

No, you just have to get off the train and wait for your next train which will arrive on the same or opposite track of the same platform (or possibly already stands there).

the system is different from you think.
trains run in mutual. i.e. subway cars go to Takatsuki in Hankyu line with no transfer, and some of Hankyu cars get into the subway line (and go to the end of this subway line). therefore, no transfer gates at Awaji station. it works as same as one company.

Thanks Uji and Ken for the prompt response, very useful!

--Emil