It is permitted to use the plastic cup and for putting in hot water there is no problem of ruining nor of cooking something hard to make it soft.
Explanation:
Gemara Shabbat 40b: R’ Yitzchak bar Avdimi said one time went with Rebbi to the bath house and I asked if I could put the container of oil inside the steaming hot bath dn eh aid take out the water with a Keli sheini and then put it in there. We learn three things from here1. Oil can be cooked. 2) Keli sheini doesn’t cook. 3) melting is considered cooking.
The Mishna in Shabbat 42a: the frying pan and pot they just took them off the fire do not put spices in side of them cut you can put it in a plate, and it is fine.”
The Gemara on 42b: ra Yosethogh that salt is like spices and in a Keli Rishon, it gets cooked but, in a Keli, sheini it doesn’t. Abaye said to hm R’ Chiya taught its like other spices that even in a Keli Sheini also cook, and this argues with R’ Nachman who says that Salt needs strong cooking like meat. Some say that Rav Yosef thought to Shulchan Aruch that salt is like other spices that it gets cooked in Ki Rishon but not in a Keli Sheini. Abaye said R’ Chiya taught salt doesn’t need cooking a Keli Rishon and this like Rav Nachman that salt cooks like a slab of meat.
Shulchan Aruch 18:9 rules like Rav Nachman. Keli rishon even after you take it off the flame it cooks as long as it is Yad Soledet Bo therefore it forbidden to put spices in, but Salt is permitted to pour it insides when e flame and salt needs cooking like meat.
Even though we rule like the second version in the Gemara for salt The Rosh ruled like the first version for spices that cooking Keli Sheini as well’ called Kalei Habishul.
This what the Rema that even though there is no “cooking after cooking” is not for dry foods but if it was baked or roasted then cooked and it forbidden to put bread into Keli Sheini if it hot.
We don’t ah comprehensive list of Kalei Habishul.
The Semag writes that we are not experts what is called Kalei Habishul therefore is good idea to stringent and not put anything in Keli sheini. “so ruled the Shemirat Shabbat.
However, in Yalkut Yosef 118:7 rules only an egg and tea are Kalei Habishul and I am forbidden put them in hot water.
What about Kalie Habishul in Keli Shlishi. The Pri megadim entertains being lenient on Kalei Habishul in Keli shlishi. So ruled R’ Moshe Feinstein.
The Aruch HaShulchan forbidden to put tealeaves in in Keli Shaini but I permitted to pour from a Keli shieni on them since that is considered Keli Shlishi.
So, Keli shlishi is permitted by many plakin.
Regarding the cup.
The Rambam writes one who melts metal or until it become red hot is toldah of mevashel. Since by heating it up he softens it. It would seem that putting the hot water in the cup is a problem.
The Shevet Halevi writes that it is permitted. Since he has no intent to cook like the Magen Avraham some things a person without intent are not anything. R’ Shlomo Zalman Auerbach says that this softening is not Bishul since it doesn’t reach the temperature, they use in melting plastic in the factories. This softening he has zero intent to make it soft, so it is permitted.
