What are the abominations that Bnei Yisrael should desist from, which would cause it to break the covenant of God. And has the covenant been broken in the past?
There are a number of verses that indicate that the covenant cannot be broken.
כִּי אֵל רַחוּם יְהוָה אֱלֹהֶיךָ, לֹא יַרְפְּךָ וְלֹא יַשְׁחִיתֶךָ; וְלֹא יִשְׁכַּח אֶת-בְּרִית אֲבֹתֶיךָ, אֲשֶׁר נִשְׁבַּע לָהֶם. (דברים ד:לא)
For the LORD thy God is a merciful God; He will not fail thee, neither destroy thee, nor forget the covenant of thy fathers which He swore unto them. (Deuteronomy 4:31)
וְאַף-גַּם-זֹאת בִּהְיוֹתָם בְּאֶרֶץ אֹיְבֵיהֶם, לֹא-מְאַסְתִּים וְלֹא-גְעַלְתִּים לְכַלֹּתָם--לְהָפֵר בְּרִיתִי, אִתָּם: כִּי אֲנִי יְהוָה, אֱלֹהֵיהֶם. (ויקרא כו:מד)
And yet for all that, when they are in the land of their enemies, I will not reject them, neither will I abhor them, to destroy them utterly, and to break My covenant with them; for I am the LORD their God. (Leviticus 26:44)
וַיֹּאמֶר אֱלֹהִים, אֲבָל שָׂרָה אִשְׁתְּךָ יֹלֶדֶת לְךָ בֵּן, וְקָרָאתָ אֶת-שְׁמוֹ, יִצְחָק; וַהֲקִמֹתִי אֶת-בְּרִיתִי אִתּוֹ לִבְרִית עוֹלָם, לְזַרְעוֹ אַחֲרָיו. (בראשית יז:יט)
And God said: 'Nay, but Sarah thy wife shall bear thee a son; and thou shalt call his name Isaac; and I will establish My covenant with him for an everlasting covenant for his seed after him. (Genesis 17:19)
This is the traditional Jewish understanding of the covenant in question.
See Exodus 19:17 which says that Bnei Yisroel were under the mountain. Rashi brings the Gemara (Shabbos 88a) and Midrash Yalkut Yisro that G-d literally lifted Har Sinai over the Jews' heads and said "if you accept the Torah, good, but if not, you will be buried there."
Why did G-d force the Bnei Yisroel to accept the covenant when they had already accepted it sight unseen in Exodus 19:8? The Midrash connects it to the law of a rape victim. Devarim 22:29 says that when a man rapes a woman, he must marry her and can never divorce her.
Accepting the covenant was a wedding, with G-d as the groom and Yisroel as the bride. Even though we agreed to marry Him, He forced us into it, so to speak, so that He would be thereby obligated to never break His covenant with us, no matter how far we might stray.