After Tawaaf, I performed two Rak‘ahs at the Standing Place of Ibraaheem (Abraham), May Allaah Exalt his mention, but with my right shoulder uncovered. Is my ‘Umrah valid? Also, at the first round of Sa`y while on As-Safa mount, I said: “Laa ilaaha illa Allaah (there is no god but Allaah)” one time and at the other rounds I repeated it thrice. Is my ‘Umrah valid?

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

Your ‘Umrah is valid and there is nothing due upon you. What you mentioned in the question has no effect on the validity of ‘Umrah.

Regarding the first issue, you should know that covering the shoulders in prayer is recommended according to the majority of scholars. It is obligatory only according to one of the two views of the Hanbali scholars who are also of the view that it is obligatory to cover only one of the two shoulders. Your left shoulder was indeed covered. Also, they are of the view that covering the shoulders is due in the obligatory prayer, not the voluntary prayer. According to the majority of scholars, the two Rak‘ahs of Tawaaf are an act of Sunnah. Consequently, the two Rak‘ahs of Tawaaf that you performed while one of your shoulders was uncovered are valid according to the views of the four Imaams.

If we supposed that the two Rak‘ahs of Tawaaf that you performed are invalid, or even if you did not pray them at all, your ‘Umrah is still valid and there is nothing due upon you. According to the preponderant opinion, the two Rak‘ahs of Tawaaf are an act of Sunnah, and this is the view of the majority of scholars, and it is the preponderant view.

Ibn Qudaamah said: "The two Rak‘ahs of Tawaaf are a confirmed act of Sunnah but not obligatory. This is also Maalik's view. Ash-Shaafi‘i has two views: one of them is that the two Rak‘ahs are obligatory because they are affiliated to Tawaaf and therefore they are obligatory like Sa‘y. As evidence of our view, however, is the Hadeeth in which the Prophet, sallallaahu ‘alayhi wa sallam, said: "Allaah Made five prayers (per day) obligatory on the servant. Whoever perfectly performs them, for him will be a promise from Allaah to enter Paradise." The two Rak‘ahs of Tawaaf are not included. Likewise, when the Bedouin asked the Prophet, sallallaahu ‘alayhi wa sallam, about the obligatory prayers, he mentioned the five prayers. Then the Bedouin asked again if there were other obligatory prayers and the Prophet, sallallaahu ‘alayhi wa sallam, answered: "No, unless you perform voluntary prayers.""

As for the second issue which is abandoning some of the prescribed Athkaar on As-Safa, there is no harm at all. According to the Sharee‘ah prescriptions, the referred to Thikr (singular of Athkaar) is said thrice on both As-Safa and Al-Marwah in all rounds. This is in compliance with what the Prophet, sallallaahu ‘alayhi wa sallam, did as stated in the long Hadeeth narrated on the authority of Jaabir describing the manner of Hajj. If a pilgrim does not ascend the mountain in the first place, there is nothing due upon him, so failing to say this Thikr is of no consequence with greater reason.

Ibn Qudaamah said: "If he (the pilgrim) did not ascend the mountain of As-Safa, there is nothing due on him. Al-Qaadhi said, 'But he is required to go through most of the distance between As-Safa and Al-Marwah.'"

We do not know of any scholar who says that it is obligatory to say this Thikr on As-Safa or Al-Marwah.

We alert that the Sunnah is to say this Thikr when one ascends As-Safa or Al-Marwah, not during Sa‘y, as your question implies.