The idea of Tashlumin comes from Gemoro Brochos 26a {Gemoro Hebrew text, Sefaria English elaboration}
….......... אמר רבי יוחנן טעה ולא התפלל ערבית מתפלל בשחרית שתים שחרית
מתפלל במנחה שתים
Rabbi Yoḥanan said: One who erred and did not recite
the evening prayer, prays in the morning prayer two Amida prayers; one
who erred and did not recite the morning prayer, prays in the
afternoon prayer two Amida prayers.
- Tashlumin is only for the shemoneh esrei. See O Ch 108 (1) MB [3] who says:
ולענין ק"ש אם יש לה תשלומין עיין לעיל סוף סימן נ"ח ובמה שכ' שם במ"ב ועי"ש עוד דברכת ק"ש אין לה תשלומין לכו"ע דהיינו אפילו אם ירצה לברך אותם בעת שקורא ק"ש פעם אחרת עם ברכותיה דלא תקנו חז"ל תשלומין אלא לתפלה שהיא בקשת הרחמים
As far as Shema if it has Tashlumin see above... and see there that the brochos of Shema do not have Tashlumin according to all opinions, that is even if one wants to say them when reading Shema another time (= after the time) with its blessings, for Chazal did not
institute Tashlumin except for “tefilloh” (= shemoneh esrei) which
is requests for mercy.
- Regarding the Tashlumin for the Shacharis shemoneh esrei O Ch 108 (1) the mechaber states:
If one erred or was forced [by circumstance] and did not pray the
morning prayer, one prays the afternoon prayer twice: the first is the
afternoon prayer, and the second as compensation.
- As far as when to say the Tefillas Tashlumin, Mishna Berurah says O Ch 108 (2) MB [11]
כי תפילת השלמה צריכה להיות בעוד שעוסק בתפילה החיובית
that it should
be at the same time when one is occupied with the prayer that is
obligatory at that time.
Halachipedia summarises from Shulchan Oruch O Ch 108,
In general, one shouldn't make any interruption in between the
original Shemonah Esrei and the Tashlumin besides for certain prayers
which are mentioned in the following halachot:
If one needs to say Tashlumin for Arvit one should say Shacharit
Shemona Esrei, Tachanun, and Ashrei and then the Tashlumin Shemona
Esrei.
If one needs to say Tashlumin for Shacharit one should say Mincha
Shemona Esrei say Ashrei and then say the Tashlumin Shemona Esreh.
If one needs to say Tashlumin for Shacharit one should wear tefillin
for mincha, both for the mincha shemona esrei as well as the tashlumin
shemona esrei.
If one needs to say Tashlumin for Mincha, according to some poskim,
one should say Arvit Shemona Esreh and Ashrei and then the Tashlumin,
while others say that one should just say Avrit Shemona Esreh and then
Tashlumin after waiting the time it takes to walk 4 amot.
This shows that it is possible to have an “Ashrei” associated with the Tashlumin but that the other prayers you mention (song of the sea every day, boruch she-amar) seem not to be part of the Tashlumin arrangement.
Of course, it is possible to say the Psalms of Pesukei Dezimro and to read and/or learn the song of the sea and the Shema at any time but this is not part of the Tashlumin arrangement.