It’s not outright wrong, but it sounds unnatural without «есть» and it changes the meaning.

This has to do with what is the main piece of new information. «У меня уже есть такой стол» means the existence of the table is the most important piece of information.

If you drop «есть», the main piece of information will be «такой». So, it will mean The table I have is already like this. This is probably not the message you're trying to convey.

Такой is used to describe masculine singular nouns in the nominative.

Такие is used to describe plural nouns in the nominative.

See here for a dictionary entry, and also consider this, a helpful guide on Russian adjectives with tables.

«Такой» is an adjectival pronoun, it works like adjectvies: so, it is normally placed before the thing it modifies. Here, такой modifies стол so it’s placed before стол.

Sometimes you can change the word order to emphasse something, but Duolingo generally doesn’t accept emphatic word order.

Est (ест, ‘he eats’) and est' (есть, ‘there is’) are different words. Also, uzhe (уже) has a voiced zh-sound, not a voiceless sh-sound.

у меня belongs together! The word by word translation is "by me". So it is not a verb as in English, where you can put an adverb between "I" and "have".

Not exactly. As a native speaker I can see a slight difference: такой practically means "the same" for one or several features. For example, you see a kitchen table in a store and want to buy it, but your friend tells you "we already have такой table". And the tables can be of different colors and shapes, but they are both for kitchen. And "similar" translates as похожий. You say that when things all in all look the same.