الحافة والتخانة كويسين بعد ما القطعة الفخارية نشفت.

Because Egyptian Arabic usually leaves out the verb to be in the present tense.

So:

الحافة والتخانة كويسين
literally = the edge/rim and the thickness good
natural English = the edge/rim and the thickness are good

This is called a nominal sentence. If you wanted a past or future meaning, you would normally use a verb such as كان or هيبقى.

Because the sentence is talking about specific features of a specific object: the edge/rim and the thickness of that pottery piece.

In Arabic, when two nouns are joined with و (and), each noun often keeps its own definite article:

• الحافة = the edge / the rim
• التخانة = the thickness

So الحافة والتخانة is exactly the edge/rim and the thickness.

Yes. التخانة means thickness in everyday Egyptian Arabic.

It comes from تخين, which means thick.

So the pattern is roughly:

• تخين = thick
• تخانة = thickness

This is very natural in spoken Egyptian. Another word you may see is السُّمك, which can also mean thickness, but تخانة is especially common in everyday speech.

Because it is describing two things together: الحافة and التخانة.

If you talked about just one of them, you would say:

• الحافة كويسة = the edge/rim is good
• التخانة كويسة = the thickness is good

But when both are grouped together, Egyptian Arabic normally uses the plural/common non-singular form:

• الحافة والتخانة كويسين

This is the natural colloquial way to say that both are good.

Because in Egyptian Arabic, the masculine plural form is very often the default plural form in everyday speech, especially with non-human things and in predicate adjectives like this.

So even though both nouns are feminine singular, the natural colloquial predicate is still:

كويسين

rather than a special feminine plural form.

This is very common and normal in spoken Egyptian Arabic.

بعد ما means after followed by a full clause.

So:

• بعد ما القطعة الفخارية نشفت = after the pottery piece dried

This is a very common Egyptian structure:

• بعد ما خلصت = after I finished
• بعد ما وصلت = after I arrived

So here, بعد ما introduces the event that happened first: the piece dried, and then its edge/rim and thickness were good.

Because the subject is القطعة, and قطعة is a feminine singular noun.

So:

• القطعة نشفت = the piece dried

If the subject were masculine singular, you would expect نشف instead.

The -ت at the end of نشفت marks the feminine singular in the past tense.

الفخارية means ceramic, pottery, or made of clay/pottery.

It comes after القطعة because adjectives normally come after the noun in Arabic.

So:

• قطعة فخارية = a pottery/ceramic piece
• القطعة الفخارية = the pottery/ceramic piece

Also, the adjective matches the noun:

• قطعة is feminine singular
• so the adjective is also feminine singular: فخارية
• and because the noun is definite (القطعة), the adjective is also definite (الفخارية)

Literally, الحافة means edge.

But in pottery context, rim may be the better English choice if you mean the top edge of a vessel or container.

So depending on the exact object, you might translate it as:

• edge
• rim

Both can be correct; rim is often more natural for pottery.

A rough pronunciation is:

el-ḥaffa we-t-takhāna kwayyisīn baʿd ma l-ʔeṭʿa l-fakhkhāreyya nishfit

A few useful notes:

• التخانة is pronounced with assimilation, so the l of الـ blends into ت: et-takhāna / it-takhāna
• كويسين is something like kwayyisīn
• نشفت is often pronounced nishfit or nishfet, depending on the speaker

So the sentence flows more like spoken Egyptian than the spelling may suggest at first glance.

Yes, that would also be natural.

Arabic often allows some flexibility in word order, especially inside subordinate clauses. So both of these are understandable and natural:

• بعد ما القطعة الفخارية نشفت
• بعد ما نشفت القطعة الفخارية

The version in your sentence puts القطعة الفخارية first, which may feel a little clearer for a learner, but both patterns are used.