Learn Hardcore Portuguese (Portugal): No fim, lavo a tigela e a tábua com esponja nova. - At the end, I wash the bowl and the cutting board with a new sponge.

bold No fim bold = at the end (of a process/sequence); neutral, common. Example: bold No fim, lavo... bold
bold Por fim bold = finally/lastly (the concluding step). Slightly more rhetorical; great in instructions: bold Por fim, lavo... bold
bold No final bold = close to bold no fim bold, a bit more formal; frequent in bold no final do dia/filme/jogo bold.
bold Ao fim (de) bold = after/at the end of a duration: bold Ao fim de uma hora, lavo... bold Only bold ao fim bold would change the meaning here.

Do I need the comma after bold No fim bold?

Yes, it’s recommended. In European Portuguese, an initial adverbial like bold No fim bold is typically followed by a comma. Without it isn’t exactly wrong when the phrase is short, but bold No fim, lavo... bold is the norm.

Why are there definite articles bold a bold in bold a tigela e a tábua bold? Could I drop them?

Portuguese uses definite articles more than English for specific, contextual items.
Here you mean the specific bowl and board you just used, so bold a tigela e a tábua bold is natural.
Dropping them (bold lavo tigela e tábua bold) sounds ungrammatical/telegraphic.
For non-specific items, use indefinites: bold uma tigela e uma tábua bold.

Why repeat the article before both nouns? Can I say bold a tigela e tábua bold?

Repeat the article. In EP you normally say bold a tigela e a tábua bold. Saying bold a tigela e tábua bold is not idiomatic.

Why is there no article in bold com esponja nova bold? Should it be bold com uma esponja nova bold?

After bold com bold, EP often allows a bare singular to express the instrument/manner: bold com esponja bold = using a sponge (as a tool).
bold com uma esponja nova bold is also correct and a bit more specific (one new sponge; it “introduces” the item).
bold com a esponja nova bold refers to a particular new sponge known to both speaker and listener.

Does placing bold nova bold before or after the noun change the meaning?

Yes:

bold esponja nova bold = a brand-new/unused sponge.
bold nova esponja bold = a new/different sponge (another one), not necessarily unused. This before/after nuance applies widely: bold um carro novo bold (brand-new) vs bold um novo carro bold (a different car).

How do I pronounce the tricky words?

bold No fim bold: [nu ˈfĩ] — nasal vowel in bold fim bold (like “feem” with the vowel nasalized, no final n sound).
bold lavo bold: [ˈlavu].
bold tigela bold: [tiˈʒɛlɐ] (bold g bold = [ʒ], as in “vision”).
bold tábua bold: [ˈtabwɐ] (two syllables: TA-bwa; stress on bold tá bold).
bold esponja bold: [ʃˈpõʒɐ] (roughly “sh-PON-zhuh,” with nasal “on”).

Do I need to say bold eu lavo bold or can I drop bold eu bold?

You can drop it. Portuguese is a pro-drop language: bold lavo bold already implies “I wash.” Use bold eu bold only for emphasis/contrast: bold No fim, eu lavo... bold (I, as opposed to someone else).

How do I replace bold a tigela e a tábua bold with a pronoun in European Portuguese?

Both are feminine; the plural direct object pronoun is bold as bold.
Neutral affirmative clause (no trigger): enclisis in EP → bold Lavo-as. bold
With negation or certain words (e.g., bold não, já, ainda, só, também, que, se bold), use proclisis → bold Não as lavo. Já as lavo. Só as lavo no fim. bold
In enclisis, hyphenate: bold lavo-as bold.

What if the objects were masculine? Any spelling changes with clitics?

Masculine forms: bold o/os bold.
After a finite verb ending in a vowel: just hyphenate → bold Lavo-o / Lavo-os. bold
After an infinitive ending in bold -r, -s, -z bold, drop that letter and change bold o/a/os/as bold → bold lo/la/los/las bold:
- bold lavar + o → lavá-lo bold
- bold fazer + as → fazê-las bold
- bold dizer + o → dizê-lo bold
With future/conditional, formal EP allows mesoclisis (bold lavá-lo-ei bold), but everyday speech prefers periphrasis: bold vou lavá-lo bold.

Is bold tábua bold alone enough to mean a cutting board?

Often yes in kitchen contexts. To be explicit, say bold tábua de cortar bold or bold tábua de cozinha bold.

Are bold tigela bold, bold taça bold, and bold bacia bold interchangeable?

bold tigela bold: bowl (mixing/eating, like cereal).
bold taça bold: cup/goblet; also a dessert bowl with a stem; also a trophy.
bold bacia bold: basin/wash-bowl; larger/deeper than bold tigela bold, often for washing. For “soup plate,” you’ll hear bold prato fundo bold.

Can I move bold com esponja nova bold earlier in the sentence?

Yes, but end-position is the default.

Neutral: bold No fim, lavo ... com esponja nova. bold
Emphasis on the instrument: bold No fim, com esponja nova, lavo ... bold
Stronger, marked emphasis: bold Com esponja nova, no fim, lavo ... bold

How do I say “at the end of X” with bold No fim bold?

Use bold no fim de + noun bold:

bold no fim do dia/semana/mês bold
bold no fim da receita/da aula bold Standing alone, bold no fim bold means “at the end (of it).”

Would anything be different in Brazilian Portuguese?

Pronoun placement: BP strongly prefers proclisis or full noun phrases; bold Eu lavo-as bold sounds unusual there. They’d say bold Eu lavo a tigela e a tábua bold; colloquially, many say bold Eu lavo elas bold (nonstandard in formal writing).
The rest (article use, adjective order, bold por fim/no fim bold) is similar, with different pronunciation.