Uttarakalamrita mentions that Lord Parshurama was born on Akshaya Tritiya.
The auspicious day known as Tretadi or beginning of the Treta Yuga, or
when the birth of Parasurama is celebrated is the 3rd lunar day in
the month of Vaisakha (April-May)-Akshaya Tritiya.
[39, Kanda 2, UTTARAKALAMRITA (OF KALIDASA)]
Yes, there is mention of firecrackers to celebrate the return of Bhagavan Rama, they were lit by fire and they would light up the skies.
Anand Ramayana, Sarga 3. 306,307
तथा कृत्रिमवृक्षांश्च पताकाश्च वजांस्तथा । वह्निसंगादोपधीनां पुष्पवृक्षविनिमितान् ॥३०६॥
No, they are not.
There are a lot of people who abstain from the use of fireworks on Diwali and celebrate the festival peacefully. There are others who use them but it's upon individuals.
This article has covered almost all the reasons covered which happened like returning of Shri Ram to Ayodhya, the appearance of Mata Lakshmi in Ksheer Sagar and much more.