Does the material matter (for example, sandalwood beads, rudraksha
beads, Tulsi beads, quartz crystal beads etc.)?
Yes, different malas are prescribed to be used for different deities. This is the topic of the 13th Patala of Sri Matrika Bheda Tantram where Mother Goddess asks Lord Shiva about the various types of beads Japamalas are made of.
Lord Shiva replies:
Vaishnave tulasi mAlA gajadantair-ganeshware |
mahamantram japed-rudrAkshamAlAyA ||
In Japa of Vishnu (mantra), Tulasi beads mala (to be used), in Ganesha
Japa beads made of elephant's tusk. Mahamantra of Kalika Japa must be
done using mala of Rudraksha beads. The wise mantra-knower must to do
Tara mantra Japa using Mahashankha mala.
Chapter 13; verses 2,3,4.
Tathaiva sakalA vidyA mahAshankhe vaset sadA || (4)
For Japa of all Mahavidyas, Mahashankha mala is apt.
Spathiki sarva-devasya pravAlaih sakalAm japet |
swarna-raupya-samudbhutAm sarva-deveshu yojitAm || (5)
Spathika mala is prescribed in Japa for all deities; the same holds true for
Coral beads mala. Malas made of gold and silver beads are also
prescribed to be used for all deities.
And, we also have:
KAlikAyAshcha sundaryA rudrAkshaih prajapet sadA |
prajapenmantram shankha-padmAkhyayoh priye || (6)
Japa of Kalika and Tripura Sundari must be done using Rudraksha malas.
For Goddess Bhairavi, use malas made of Shankha and lotus seeds.
(A partial answer for the time being. I may update it with more info about the other queries)