I have started learning Japanese some time ago, but I still can't understand the sutras and chants that are used in Zen ceremonies, such as Takkesa Ge, Maka Hannya Haramita Shingyo, Fu Eku, Jin Ho San Shi etc. I understand it may be in a more archaic form, so how well can Japanese speakers understand these texts?
These are the Sino-Japanese readings of Chinese characters, so the grammar isn't even that of Japanese, but of Chinese. This can be seen by the fact that the verb (行, 度) precedes the object (深般若波羅蜜多, 一切苦厄) in each clause:
Japanese has SOV (subject-object-verb) word order, in contrast to Chinese's SVO.
As for whether the sutras are intelligible by ordinary native speakers, even aside from the differences in syntax, my assumption would be "no" unless they are familiar with the glosses. As an example, even though the Chinese character 度 is also present in Japanese, note that in Japanese, it only has the rare reading 度る with the definition 推し量る "to surmise", and not the specialised Buddhist sense of "to save from purgatory".