Should one say the Adhan and recite aloud when praying alone?

In Fatawa al-Lajnah al-Daa’imah wa’l-Buhooth (6/392) it says the following:

It was reported that the Prophet (peace and blessings of Allah be upon him) used to recite aloud in the two Rak’ahs of Subh (Fajr), and in the first two rak’ahs of Maghrib and ‘Ishaa’. So reciting aloud in these cases is Sunnah, and it is prescribed for his Ummah to follow his example, because Allah says (interpretation of the meaning):

“Indeed in the Messenger of Allah you have a good example to follow for him who hopes in (the Meeting with) Allah and the Last Day and remembers Allah much.” [al-Ahzaab 33:21].

It was reported that the Prophet (peace and blessings of Allah be upon him) said: “Pray as you have seen me praying.”

Reciting silently when one is supposed to recite aloud implies that one is neglecting the Sunnah, even though this does not invalidate the prayer.

So the Muslim should recite aloud during Fajr prayer and in the first two rak’ahs of Maghrib and ‘Ishaa’, even if he is praying alone, because this is the Sunnah.

He should not recite aloud during Zuhr and ‘Asr. He should recite aloud during the first two rak’ahs of Maghrib and ‘Ishaa’, and in the fard of Fajr prayer, and he should recite silently in the third Rak’ah of Maghrib and in the last two rak’ahs of ‘Ishaa’. These times of reciting aloud or silently are mustahabb when praying alone, according to Imaam al-Shaafa’i and others. The Hanbalis say: the person who is praying alone has the choice of reciting aloud at the appropriate points; if he wants to he may recite aloud, and if he wants to he may lower his voice.

(al-Mufassal li Ahkaam al-Mar’ah by ‘Abd al-Kareem Zaydaan, p. 330).

With regard to the second part of your question, in Fatawa al-Lajnah al-Daa’imah wa’l-Buhooth (6/56) its says:

It is permissible for a person praying alone to pray without the Adhan, but if he is in the desert or on a remote farm and the like, it is prescribed for him to recite the Adhan even if he is going to pray alone, and the Iqaamah is prescribed at all times, because of the general meaning of the evidence, and because Abu Sa’eed al-Khudri said to ‘Abd-Allah al-Ansaari: “I see that you like sheep and the wilderness. When you are with your sheep in your wilderness, and you recite the Adhan for your prayers, raise your voice, for no jinn or man or anything within the voice range of the muezzin hears the call, but will bear witness for him on the Day of Resurrection.” Abu Sa’eed said: I heard this from the Messenger of Allah (peace and blessings of Allah be upon him).

(Narrated by al-Bukhari, 1/151).

And Allah is the Source of strength. May Allah bless our Prophet Muhammad and his family and companions, and grant them peace.