Have you heard of concepts called "gender" and "the third declension"?
According to some studies on teaching Modern Greek that I came across a couple of years ago, these are the type of mistakes that 7-9 year olds might make. You can write like this with your free-writing brain, but then go back over it with your analytical brain to check that. Eventually, you are aiming to compose in your mind, then check it in your mind, then write the corrected form down. In other words, the aim is to internalise, as much as possible, the filtering.
I respect the fact that you have thrived and become an independent self-directed learner in the time since when we were more paw in glove while Lysias and the other works. I hope that you would allow me to say some superficial and obvious things about what you have written.
Wes Wood wrote:παίδιον τις καὶ ὁ πατρὸς αὐτοῦ βλέπουσιν ἀκρίδον καταβαίνοντα εἰς φραγμὸν. βλέψας, τὸ παιδίον τρέχει εἰς τὸν φραγμὸν καὶ ψηλαφῶν εὑρίσκει αὐτόν. εὑρὼν αἰρεῖ τὸν ἀκρίδον ὑπέρ τὴν κεφαλὴν συμβιβάζειν τῷ πατρὶ ὅτι εὗρεν αὐτόν.
παίδιον τις - There is a difference in gender here.
ὁ πατρὸς - one of these is nominative, and the other genitive.
ἀκρίδον - somebody could write a PhD on it if this form turned up in an ancient manuscript.
καταβαίνοντα - the case here is correct, but the gender is not.
βλέψας, τὸ παιδίον - the gender doesn't match here
τρέχει εἰς τὸν φραγμὸν - that seems like an unwise thing to do, it might be prickly.
αὐτόν - gender
τὸν ἀκρίδον - case and declension
ὑπέρ τὴν κεφαλὴν - are you trying to say "by the top of the head"?
συμβιβάζειν - "prove" in the sense of the (English) word that you are trying to use means δηλόω ("let me show you and you can see for yourself"). συμβιβάζειν means prove bring several lines of reasoning together to make a point constructed on them (QED).
There is a lack of conjunctions here. Think in terms of "no punctuation". Conjunctions are needed to break up the text. In a descriptive text like this, thing more or less flow, so words that mean "kand" or "next" might do.