Should Stotras, such as Ganpati Stotra, Durga Stotra, etc., and Mantras be recited aloud or can they be recited just in mind? Also, does the absolute correct pronunciation of words in Stotras matter?
Stotras should never be recited in mind. In Stotras' case there should be loud recital.
In Mantra Japa, the best form is Manasa Japa (the silent chant). Loud Mantra Japa is considered as the inferior kind of Japa.
Manasaa yah smaret stotram vachashaa vaa manum japet |
Ubhayam nishphalam devi bhinna-bhaandokadam yathaa ||
O Devi! The act of reciting Stotras in mind and the act of repeating Mantras loudly are both fruitless just as is the act of storing water in a pot which has a hole in it.
Kularanava Tantram 15.57
For Mantras, the rule is exactly the opposite as already shown in this answer.
Ucchair japohadhamah prokta upaanshur madhyamah smritah |
Uttamo mAnaso devi trividhah kathitah japah ||
O Goddess, the loud japa is considered as the worst kind, the whispering japa (upanshu) is the middling kind and the japa that is done completely in the mind (maanasa), is the best form of japa. These three are said to be the kinds of japa.
Kularnava Tantram 15.55
That's why Stotra recital must always be done loudly.
The rules for reciting Mantras are many many and all of them can not be covered in one answer. In comparison, rules of Stotra recital are only a few and simple to follow.
The following verses are from the Varaahi Tantram, quoted in both Tantrasaarah and Ahnik Krityam:
Pranavanchaadime datvaa stotram vaa samhitaam pathet |
ante cha pranavam pranavm dadyaadi-tyuvaachaadi-purushah ||
Stotre cha samhitaayaancha shlokamantyam dviruccharet |
Manasaa na smaret pathedakaagramaanasah ||
One should chant OM before and after reciting a Stotra or a Samhitaa. And, one should never recite Stotra in mind; one should recite it loudly with a concentrated mind.
According to the Stavakusumajnali published by Nagpur Ramakrishna Math
stotras are part of vachik (vocal) puja (worship) called 'Bangmoyee puja' and should be chanted with correct chhanda and pronunciation and audible clear voice, understanding the meanings of the stotras and contemplating on the 'bhava' (page 4-5).
For those who are are able to pronounce correctly but have devotion, the scripture says
murkho vadati vishnaya dhiro vadati vishnave/ubhayos tulyam artham cha bhavagrAhi janArdanah (Narada-pancharatra)
meaning that the uneducated one says visnaya and the learned says vishnave.To God both are the same as He accepts the devotion alone.
Samarth Ramdas swami says, you should say name of lord ram loudly
Jani Bhojani Naam Vache vadave ATI Adare Gadya ghoshe mhanave Harichintanene ann jewit jave Tari shri hari pavijeto swabhave
While wandering in the masses, and while consuming food, recite the name of lord Ram Recite it with respect, clarity and loudly You should consume your food while thinking about God By following this you can attain god
Saarth Manache Shlok, Shlok #89