I often see mesoamericana pre-Columbian societies referred to as “Stone Age” though I’ve also heard of instances of metalworking. Does this metalworking extend to tools such as axes and machetes? If not, what kind of rock would their society use for axe heads; granite, tuff, flint, et cetera? Or was their method of dealing with the jungles and bush land just to burn it?
First off, whoever is referring to any pre-Columbian society as "Stone Age" is wrong. The term Stone Age is used to refer to a specific period of time in Eurasian chronology based off of specific characteristics unique to the archaeological record of the area in question (the start and end of Stone Age in one region does not always match with another's). The term Stone Age didn't gain in popularity until after Christian Jurgensen Thomsen (1788-1865) used the term to refer to one of five periods of time in the archaeological record of Denmark. Even in Thomsen's time in the early 1800s, he knew that bronze and stone tools were still being produced in the Iron Age. His classification system was reliant not on the presence or absence of certain technologies in different Ages, but on differences in the material he found. His five age system differentiated between stone artifacts being produced in the Stone Age vs. the Bronze Age vs. the Iron Age as well as differentiating the bronze artifacts from the Bronze Age vs. the Iron Age (Trigger 2010: 121-127). So even in Eurasia the labels of Stone Age, Bronze Age, and Iron Age do not reflect a level of technology, but rather the style, form, and characteristics of the material culture in these different periods of time.
As for the Maya and their tools, they would have used basalt cobbles with one end ground to an edge as their tree cutting implement. While stone tools would have taken a little longer to cut down a tree (approximately a 1:4.4 ratio), it still did not take much time to cut down a tree. A steel machete can cut down a 4 cm diameter banderio in 3 minutes, 8 cm pine in 10 minutes, a 10 cm sauce in 15 minutes, and a 13 cm cacao menace in 20 minutes. You can calculate how much time stone axe would need to cut down a tree by using the energetics formula T = 2.3d^3 x 1.3h where d = diameter of the tree in square feet and h = specific density of the tree in question (Abrams 1984: 157-158).
Abrams, Elliot Marc.
1984 Systems of Labor Organization in the Late Classic Copan, Honduras: the Energetics of Construction. Ph.D. dissertation, Department of Anthropology, The Pennsylvania State University, State College.
Trigger, Bruce G.
2010 A history of archaeological thought. Cambridge University Press