Does the prayer become invalid if the worshipper performs actions unrelated to it during the Friday sermon, such as applying perfume and distributing it to those next to him? And what should the worshipper do if the congregational prayer begins while he is in the first rak'ah of Tahiyat al-Masjid (prayer for greeting the mosque)?

Whoever attends the Friday prayer must listen attentively to the sermon, and it is not permissible for him to distract himself from it with anything. Distributing perfume to the worshippers during the sermon distracts them from it, and it is similar to touching pebbles, which is prohibited in the Hadith. Speaking and fidgeting are forbidden from the moment the preacher begins the sermon until he finishes it. So, there is no harm in distributing perfume before the sermon begins, but it must be stopped as soon as the Imam starts it. If the prayer is established while you are in the first rak'ah of tahiyat al-masjid (greeting the mosque prayer) or the regular sunnah prayer, you must interrupt your prayer and join the congregational prayer.