The answer to your three original questions are
: yes, unless you go outside the ticket gate on the way.
It is only when you go without seat reservation
that the Haruka discount ticket covers the Limited Express surcharge.
When you head to Kyobashi Station (in Osaka City),
you don't have to travel as far as to Shin-Osaka Station.
In Tennoji Station
you can get off your Haruka train
(arriving at Track 18)
and catch a train on the Osaka Loop Line's Inner Loop
(departing at Track 11 or 12),
which brings you to Kyobashi Station in around 15 minutes.
- JR-West: Map (in Japanese) of Tennoji Station
https://www.jr-odekake.net/eki/img/premises/0620831.pdf- JR-West: ICOCA & HARUKA
https://www.westjr.co.jp/global/en/ticket/icoca-haruka/Your first / second discount ticket
is a one-day one-way ticket
which lets you travel (by non-Shinkansen lines)
from / to Kansai-airport Station (in KIX Terminal 1)
to / from a station in the Osaka City Zone,
on the conditions
that you can take a Haruka train without seat reservation
to / from the designated station (among Tennoji and Shin-Osaka)
and that you cannot make a stopover outside the ticket gate.
When you exit JR through a ticket gate, the ticket is collected.
You can choose different stations in the Osaka City Zone
to get to and to get back from.
