It means: "Concerning this problem, which it perforce fell to me to unravel, it is appropriate to report the following solutions publicly; Preparation for the solution."
(1) I bolded the part of the Latin that you are asking about in particular.
(2) Sequentes does not modify problematis. It is either nominative or accusative and in this context either just means "the next things", but in English we can render it like an adverb, "Next". Euler is using this word to introduce what is coming next, the next things he will be discussing. Or, as Kingshorsey comments below, it might modify solutiones, "the following solutions".
(3) Ergo can act as a conjunction or a preposition. In this context, when it is used with the genitive, it is more like a preposition.
(4) Quidem has a context-sensitive meaning that generally intensifies whatever is being discussed so it is often translated as "indeed". Here, in a mathematical context, the implication is that he was required for some reason to solve these particular problems I guess as a precondition for solving some other problems.
(5) in medium refers to what is in the community or what is public; visum est means what is seemly or right or appropriate
(6) I am not exactly sure why solutiones is plural. The full context involves I believe a single solution to the problem, so why Fermat is using the plural is not clear to me.