What is the approximate limit for the "long period" during which the intention for prayer or ghusl should not precede them? And how is the heartfelt intention made for Fajr prayer without uttering it, and is this done by recalling the name of the prayer?

Intention (niyyah) is a pillar of prayer (Salah). It is best for it to be simultaneous with the takbirat al-ihram (the opening 'Allahu Akbar'). However, it is permissible for it to precede the takbirat al-ihram by a short time, which is determined by custom and common practice. Imam al-Haramayn, al-Ghazali, and others held that a customary simultaneity, which is considered a preparation for prayer, suffices, and this is the preferred view. The place of intention is the heart, and it is not legislated to utter it verbally. It is sufficient to be mindful of the intention to perform the prayer and specify it.