For example, why is it shown in Wiktionary that the etymology of such words like eat, есть （which means eat in Russian）comes from 3-rd person singular form *h₁édti in PIE? Are they really related or I misunderstood something?
The usual dictionary form of PIE verbs is the 3rd person singular, and in this case the implication that the forms are fully cognate is not intended. The infinitive ending reconstructed for Proto-Slavic, *-ti, is traditionally taken to be a locative ending (PIE *-tēi̯) originally, and its resemblance to the PIE 3rd person singular ending *-ti is accidental.
(PIE itself didn't have any infinitives, so infinitives in daughter languages are necessarily innovations.)