Regarding your question:
what is the realtionship beteween the substantial form of a bodily being and its act of existing, ie. its esse?
It seems that Aquinas says that form produces/gives esse ( "...beacuse form causes existance in act,..."), but how can form give esse if the form already needs to exist in order to give something?
From my POV, Aquinas's above writing may be better understood following theory of Forms in Plato's dialogues and general speech. According to this theory, the physical realm is only a shadow or image of the true reality of the Realm of Forms. The Forms are abstract, perfect, unchanging concepts or ideals that transcend time and spaces. The theory is considered to be a classical solution to the problem of universals. Most western philosophers followed this Greek tradition to essentially propose "Form precedes essense/existence (a being's act of existing in your book's language)" metaphysically. Of course, Aquinas may actually thought independently as @Conifold commented below, but I found Platonism is easier for me to understand why Aquinas says "form causes essense", thus nothing super inexplicable here under this view. Also Cardinal Nicholas of Cusa applied similar writings in his On Learned Ignorance which employed the word "quiddity", "qua", etc. He used "enfolding" referring form, and "unfolding" referring existence in itself, since enfolding precedes and thus causes unfolding, you may also use this metaphor to help you understand Aquinas's meaning since Nicholas of Cusa fairly admired Aquinas and was very well versed in Aquinas' writings.
Of course there're different schools of thought like traditional Nominalism (Ockham's razor), Skepticism, and postmodern Existentialism as summarized by Sartre's famous proposition "Existence precedes essence" which replaces above form with existence, and in the meanwhile separates a being's existence from its essense.