1- Recap of the Halachos of making tea and coffee on Shabbos:
Two ways of making leaves according to the Mishna Berurah (pouring water on the leaves; tea sense).
Cold sense vs. warm sense.
Sense into the water vs. water into the sense as it pertains to the coloring of the water (seemingly only an issue of appearance and not מעיקר הדין).
Tea in a Kli Shlishi (Aruch HaShulchan vs. Rav Moshe).
Instant coffee- possible position of the Chazon Ish (based on the testimony of Rav Chaim Kanievsky on how he made tea) that water poured directly from an urn retains the status of a Kli Rishon.
Solids that dissolve (if they have the status of a liquid or a solid as it pertains to the Halachos of בישול אחר בישול).
The difference between salt and fats (instant dissolve vs dissolving over time) and the implications regarding instant coffee.
Various stringency’s to consider for making instant coffee due to various potential issues (בישול אחר אפייה אחר בישול; since it instantly dissolves it can potentially have the status of a liquid which requires a Kli Sheini; נולד; coloring of the water).
2- Practical Halacha for instant coffee:
Rav Moshe- מעיקר הדין you can pour directly from a Kli Rishon into a Kli Sheini for instant coffee, but states that he himself is stringent to use a Kli Shlishi but does not say why.
Possible reasons can be that he considered some of the above concerns:
For the above concern of instant coffee being a liquid, Kli Sheini would suffice; for the above concern of בישול אחר אפייה אחר בישול, one would require a Kli Shlishi (Rav Eliyashiv). For the above concern of the Chazon Ish with regards to pouring from an urn, a Kli Shlishi would essentially be considered a Kli Sheini.
Another possible explanation in Rav Moshe from Rabbi Goldstein that perhaps one day they will not pre roast the coffee (and why that’s not likely since Rav Moshe didn’t recommend this stringency on anyone else but himself).
3- Brewed coffee:
Kli Sheini would not work despite it being pre-roasted since יש בישול אחר צלי.
Kli Shlishi-
Rabbi Felder- despite this being comparable to tea, the above position of the Aruch HaShulchan regarding tea in a Kli Shlishi (that it is prohibited), only applies to what he said it by (tea) and not brewed coffee.
However the Mishna Berurah does equate coffee and tea so it should seemingly be an issue by brewed coffee as well.
Advantage of brewed coffee over tea since it is pre-roasted and there are Rishonim that hold אין בישול אחר צלע. (However this maybe different since even following the roast it is inedible).
4- Cone (single) drip coffee:
This would fall into the above discussion despite there seemingly being an issue of בורר since the position of the Shulchan Aruch is that בורר is only when you separate a mixture vs. here where you start and end without a mixture (despite there being a mixture in between).
5- instant noodle soups:
Multiple elements:
Noodles, spices, vegetables
Noodles- often times deep fried and therefore considered cooked and those would be permitted since אין בישול אחר בישול by a דבר יבש. However, many are steam cooked and then dehydrated which is not the equivalent of cooked in water which can be an issue even in a Kli Sheini.
Vegetables- dehydrated and not cooked so that would definitely be an issue even in a Kli Sheini. (Some companies have it in a seperate packet so you can just avoid).
Spices- מעיקר הדין should be fine. However there are multiple considerations; these spices were not around in the times of Chazal, finely ground according to Rav Shlomo Zalman.
6- Instant noodle soup in a Kli Shlishi-
According to Rav Moshe and the above Rav Felder and many others it would be okay.