Is the merit and reward of prayer mentioned in the Hadiths, such as "Whoever prays Fajr is under the protection of Allah" and "Whoever prays the two cool prayers (Fajr and Asr) will enter Paradise," intended for praying in congregation at the mosque or praying at home?

Performing Fajr prayer in congregation places the Muslim under Allah's protection, and the hadith (definition) "Whoever prays the two cool prayers (Fajr and Asr) will enter Paradise" does not restrict this to congregational prayer. As for the hadith, "Whoever misses the Asr prayer, his deeds will be nullified," some scholars interpreted it as not performing it in congregation. Al-Ayni, in 'Umdat al-Qari, mentioned other interpretations, including that "missing it" refers to not praying it within its preferred time, or missing it by sunset, or by the sun's yellowing, or missing it due to forgetfulness, or intentionally, or missing it in congregation. A prayer performed individually at home, even with the ability to pray in congregation, is valid, though sinful and entails missing out on much good. If one prays in congregation at home, they will receive the reward for it, but will miss the virtue of praying in the mosque.