I understand the question as follows: Do all (or a notable majority of) languages "with cases"
make use of at least one construction for bivalent verbs where the non-subject argument is a noun phrase in a case other than the accusative, in addition to making use of the construction where the non-subject argument is a noun phrase in the accusative case (or in whatever case is normally assigned to the O argument of a transitive verb)
have some bivalent verbs that are usually used with an accusative argument, and other bivalent verbs that are usually used with a non-accusative argument, such that for any language X with cases Nom (or Erg), A=Acc (or Abs), B, C..., it makes sense to make lists of "bivalent verbs in language X that take a noun phrase in case A as the non-subject argument" vs. "bivalent verbs in language X that take a noun phrase in case B, C... (etc.) as the non-subject argument"
(The first but not the second bullet point could be true in a language where the case of verb arguments was determined solely by syntactic rather than lexical factors; e.g. a language where any bivalent verb with a noun phrase as its complement takes a genitive argument when negated and an accusative argument otherwise.)
The question is not asking about the existence of verbs that can be used in both constructions, or how the distribution of different constructions among different verbs varies between languages.
I'm using the phrasing "non-subject argument" because, as mentioned in the comments, it can be debated whether it is correct to refer to the non-accusative noun phrases in these constructions as "(direct) objects"; that is also why I am using the phrase "bivalent verbs", since it seems to be debated whether the verb in such constructions can be called "transitive".
Bivalent and divalent are synonyms meaning "possessing a valence/valency of two." That is, a bivalent/divalent verb is a verb that has two arguments. (In the cases that you are interested in, both arguments are noun phrases, but bivalent verbs can also take phrases of other categories as arguments: e.g. in the English sentence "I believe in him," "believe" is a bivalent verb with the following two arguments: the noun phrase/NP subject "I" and the prepositional phrase/PP complement "in him".)
In your German example sentences "Ich sehe dich" and "Ich helfe dir", "sehe" and "helfe" are bivalent verbs that take as their non-subject arguments noun phrases of two different cases (accusative and dative respectively). (The verb sehen can also be used intransitively, but that is not relevant to your question.)
I don't know; it might depend on the case inventory of the language and on how we define "case" and "language with cases"
I think it may depend on what you mean by "case". I don't think it is a very clear term, and it is sometimes used to refer to a morphological category but sometimes to a syntactic category.
There seem to be a lot of different things that can potentially be encoded by nominal inflection, and not all inflected forms can necessarily be used as the complement to a verb.
One example, even though it isn't a typical or central example of a "case", is the English -'(s) "genitive" construction. As far as I know, no finite verb in English takes an argument marked with the -'(s) genitive: this marking is used only in other grammatical contexts (mainly on noun phrases used as determiners; also on noun modifiers in the "classifying genitive" construction, on certain arguments of non-finite verb forms in certain contexts, and arguably on the complement of the preposition of in the "a friend of mine" construction).
The English -'(s) construction is difficult to analyze morphologically, which is part of the reason why there is argument about whether it is a case. But if it is considered a case, then you might be able to say that English is a language where a noun phrase used as the complement of a bivalent verb can only be in one case (the one traditionally referred to as the "accusative case", although some modern linguists seem to be reluctant to apply that traditional categorization to the modern English case system*). (For the purposes of this argument, we need to ignore the use of forms like "I" (i.e. forms traditionally called "nominative") in contexts like "It is I" or "They already told my parents and I"; some linguists argue that these uses of "I" do not actually belong to the core syntax of English, but instead are used and found acceptable by speakers of standard English for some other reason.)
I don't know of a clearer counterexample against your examples of Turkish, German, Russian, Greek and Latin, but I am not familiar with the variety of case systems used across the world. Like you, I have the impression that languages with cases frequently have bivalent verbs that select arguments of a particular case other than the main case used for objects.
*For example, in some theories, "nominative" is redefined or reinterpreted as the absence of case; based on this, Omer Preminger says in "Case in 2017: some thoughts" that "insofar as English has anything you’d want to call ‘nominative’ [...] it’s [...] the thing we’ve been calling ‘accusative’ or ‘objective’ case" and describes what is traditionally called the "nominative" in English as instead being a case "assigned by T0 under closest-c-command" (page 29).