Bay leaves: yes
Chili paste: no
I was born and raised in the Philippines and only came to the States about eight years ago. I have eaten and cooked a lot of adobo in my time. Family, friends, restaurants, street vendors, everyone uses bay leaves. I have never encountered chili paste in adobo.
Adobo is an all-meat dish, which is a rarity in Filipino cuisine. Traditionally, Filipino dishes tend to feature fish or vegetables. Dishes that have meat as the main ingredient tend to have been introduced from elsewhere: lechon, afritada, menudo, kare-kare. Notice the names? Filipinos will happily adapt a disk, keep the name and adapt it to suit their tastes. Other dishes that have been adapted include pancit, mami, bihon, sotanghon, siopao, bistik, lugao, arroz caldo.
I’ve traveled (granted, not too extensively) throughout Southeast Asia and have not encountered any other food like Philippine adobo. Given that, the prominence of meat in the dish, the similarity of the ingredients to the Latin American marinade, and the name itself, I’d say that adobo was introduced or developed during the Spanish colonial period.