As salam alaykum, Since I got pregnant, various people have been advising me to recite Surah Muryum often as it eases the labor. Is there any basis for this assertion? Should I start reciting Surah Muryum frequently? I`ve also heard that the dua` made during the labor (during the pain) is accepted. Is this true? Also, are there any dua`s etc. that we can make to make the labor easy and less painful? Are there any general dua`s from Sunnah that I can make use of for stress/anxiety? Jazakum Allah Khayraa, Was salaam

All praise be to Allah, and may his peace and blessings be on the last and best prophet and messenger, Muhammad.
There is no proof on that particular effect of surat Maryam. However you may use any part of the Quran as incantation, including surat Maryam. The reason why people associated that with surat Maryam is because it mentions the pregnancy and child birth of Maryam.
The Du’a’ of people in distress is more likely to be accepted by Allah who is near those in pain and disease. He says:
أَمَّن يُجِيبُ الْمُضْطَرَّ إِذَا دَعَاهُ وَيَكْشِفُ السُّوءَ وَيَجْعَلُكُمْ خُلَفَاء الْأَرْضِ أَإِلَهٌ مَّعَ اللَّهِ قَلِيلاً مَّا تَذَكَّرُونَ”
“Or, Who listens to the (soul) distressed when it calls on Him, and Who relieves its suffering, and makes you (mankind) inheritors of the earth? (Can there be another) god besides Allah. Little it is that ye heed!” (27/62)
You may say the following supplications
” اللهم لا سهل إلا ما جعلته سهلاً وأنت تجعل الحزن إذا شئت سهلاً ”
“O Allah nothing is easy unless you make it so, and it is you who turns hardships into ease.”
If you are hurting, you may say the following while touching the site of pain:
“بسم الله، ثلاثاً وقولي سبع مرات: أعوذ بالله وقدرته من شر ما أجد وأحاذر”
“In the name of Allah (three times) and I seek refuge in Allah, his might and power from that which I suffer and that which I fear.”
Incantations are permissible as long as you use the Quran and authentic sunnah. The best verses and chapters of the Quran in this regard would be: al-Fatihah (1) and al-Mu’awedhatain (113) & (114). You may also repeat ayat al-Kursi (2/255).
May Allah make it easy for you and grant you a good progeny who will be a pleasure for your eyes and comfort for your heart.
Allah knows best.