For example, is the following paragraph correct?
Le week-end dernier j'ai joué au foot avec mes amis. Ensuite je mangeai le dîner.
Does this mixing of the two tenses work? If not, what tense would be better in this case?
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That is not correct; you must write it as shown below; you must use the "passé composé" again. There is no other way.
Le week-end dernier j'ai joué au foot avec mes amis. Ensuite, j'ai diné.
The "passé simple" can be used in both sentences and all is fine; however this can't be used in the spoken language; this use of tense is found only in the literature; it can't be used either in common sorts of writing, letters or e-mails to friends, newspaper articles, and the like.
An important point due to jlliagre as a comment should be mentionned; most likely, the context of a diner being eaten after playing is not, for example, that of a usual happening, when diner is provided by some organisation, or sponsor, or well intended individual; that aspect would have to be made unambiguous, anyway, by some sentence before the two sentences given. As, very probably, the two sentences given are not construed as pertaining to such a context, the form "je mangeai le diner" (which appears then to be a mere literal transcription of the English "I ate diner") is not proper; it is unidiomatic in French; instead, one must say "Ensuite, je dinai.". Since, in the light of the tense in the sentence that precedes, the tense used (passé simple) does not match, the idiomatic form in the right tense has to be "Ensuite, j'ai diné.".
As a consequence of the 1990 reform of the French language and thereby of its simplification you do not have to use a circumflex accent on the i of "diner"; more precisely, you have the possibility to chose between the traditionnal spelling and the new; there are a few other words that have been simplified in this manner and you might want to look into that (paraitre, apparaitre, gout, faite (sommet), gite,…). There are exceptions though; "âge" has been simplified (now you can write "age") but a good number of others haven't (château, hâte, tâche,…). (In my opinion the simplification hasn't been carried out far enough; let's hope that before long appears a resurgence of the impetus for innovation in the language so as to make complete the good work that has been started!)