There is no country where I felt safer than Chile.
For Chile, I rode a mountain bike with smooth inverted tread tires. That worked very well for my riding which was mostly on paved roads, but included 2 crossings of the Andes on unpaved roads. BTW, the passes between Chile & Argentina in the lake region are considerably lower than those further north. Go tohttp://www.turistel.cl/ to see some excellent maps. Click on "mapas ruteros", then on individual letters to open the maps. It looks like there have been some additional roads paved since I was there 10 years ago. You could probably use a touring bike if you intend to stick to paved roads most of the time.
Hospedajes (B&B) were easy to find every night. However, my trip was in December, and supposedly few Chileans go away on holiday before January 1. Therefore, it might be harder to find a vacant room in January.
The Yucatan of Mexico is another incredibly safe area. The Mayan culture is alive and well in small villages and big cities alike in the Yucatan, and I always felt at ease everywhere I went. Female cyclists have said the same thing about the Yucatan. The safety situation varies elsewhere in Mexico. Obviously there are problems in much of the US/Mexico border area, and Mexico City (though I personally had no problems at all in Mexico City, and really liked the city). My friend & I felt quite safe biking in Michoacan and Jalisco states. I've been on 2 non-biking trips elsewhere in Mexico (Zacatecas, Guanajuato, Queretaro, Puebla, Oaxaca states & Mexico City) and had zero problems and felt I had to remain somewhat vigilant only in Mexico City. I have been told one first-hand story by a cyclist who was robbed in a rural lowland Pacific coastal area, and read a similar first-hand story by a cyclist on the central Mexico Gulf coast. (Personally, the only country where I was ever in danger on my bike trips was, of all places, Switzerland.)
Distances between towns can occasionally be great in Mexico, so you do have to take some care that you can find a room for the night, but I never had a problem myself. One final thing, it's extremely easy to bring a bike with you when you take buses in Mexico & Chile.