In everyday speech, these are generally fine.

• vingt can sound as if it has /ɑ̃/ like “an” rather than /ɛ̃/ like “in” depending on how far back the speaker pronounces that vowel. Compare the first two recordings here. And here, one speaker actually uses the same vowel for “cent” and “vingt”, whereas the other doesn’t.
  • To my knowledge, most speakers delete /t/, but some do not. However, in “vingt-et-un”, the liaison from “et” means you will hear the /t/ on the end of “vingt”. Listen here.
• et can be virtually elided in fast speech — not totally but enough to make it hard to hear
• quatre will tend to lose the /r/, much like every word ending in “-re” or “-le” (table /tab/) when speaking informally

We’d need to see the video to know for sure, but from the description that pronunciation is okay.

I “Quatre” will be pronounced “cat” especially in rapid speech. At the end of a sentence it is never pronounced “cat” (substandard).

• Combien de chaises se trouvent dans cette pièce ? Il y en a quatre.
• Combien en voyez-vous ? J’en vois quatre. Je vais les prendre tous les quatre.

For instance, it is impossible to say “quatre cents fois” rapidly, you have to say “cat cents fois”. However, when speaking slowly you do have a choice and it is not wrong to say “quatre cents fois” in full.

II “Vingt-et-un” should always be pronounced /vɛ̃.t‿e œ̃/. The type of pronouciation that you can hear on this page, https://fr.wiktionary.org/wiki/vingt-et-un#Prononciation, is not recommanded; you can still make out /e/ but it is blurred and almost melted into the following /œ̃/; that is not typical of people who speak well.

III “Vingt” is pronounced “vint” in some régions of France, but that is not standard French; the standard pronunciation is “vin” [vε ̃] (TLFi); here are three very good pronunciations of “vingt”: https://fr.forvo.com/word/vingt/.

Quatre is often pronounced a little like the English “cat” or “cut” (but with a slightly different vowel) unless it is followed by a word starting with a vowel in which case the /r/ is almost always pronounced (e.g. Elle a quatre ans). The final r might not be pronounced in quatre euros (c’est quat’euros) and in a very few cases, an extra /z/ appears in spoken French (quatre-z-yeux, quatre-z-amis, …). See La fausse liaison dans "quatres enfants" ?

When quatre is ending a sentence, it is also often pronounced /kat/ (e.g. J’en ai vu quat ^re.)

Vingt before a pause can be pronounced with or without the final t. There is no single “standard pronunciation”, the standard mostly depending on the region. The g is on the other hand never pronounced. The pronunciation of the nasal vowel in widely vary depending on the region or the people, and might indeed be close to the vowel /a/. The et of vingt-et-un is always pronounced so vingt-et-un is never pronounced like would be the hypothetical vingt-un (or vingt ans) and even less like we pronounce quatre-vingt-un (where no /t/ is heard).

See also: Pronunciation of "vingt"

and

Mathieu Avanzi, le français de nos régions :

La langue populaire, à peu près partout, réduit au premier élément les
groupes consonantiques finaux dont le deuxième élément est r ou l
: quatre prononcé °[kᴀt] ; souffle prononcé °[suf].
_{À
l’intérieur d’un syntagme, cela appartient simplement au registre
familier : Votre papa [vɔt pᴀpᴀ]. Dans le nom composé
quatre-quatre (§ 597, d, 1°[« véhicule automobile dont les quatre roues sont motrices. »]), la prononciation [kᴀt] pour le premier
élément est à peu près générale.
Devant voyelle,
le groupe reste intact : Quatre amies.
Autres réductions
populaires : -isme, -iste prononcés °[is] dans communisme,
communiste, par ex. ; — ex- prononcé °[ɛs] devant consonne : dans
exclure, par ex.}

[ Le bon usage, Grevisse et Goosse, éd. Duculot, 14^e, §36
« suites consonantiques » c) ]