هذا المتجر رخيص، لكنني لا أذهب إليه كل يوم.

هذا is a demonstrative (this) and it typically comes before the noun in Modern Standard Arabic: هذا المتجر = this store.
You can also see a post-nominal demonstrative in some contexts (especially in certain styles/dialects), but in MSA the normal, neutral order is demonstrative + noun.

In Arabic, present-tense “to be” is usually not expressed. So هذا المتجر رخيص is a nominal sentence meaning This store (is) cheap.
If you wanted past tense, you would use كان: كان هذا المتجر رخيصًا = This store was cheap.

Adjectives agree with the noun in gender. المتجر (store) is grammatically masculine, so the adjective is masculine: رخيص.
If the noun were feminine (e.g., هذه البقالة رخيصة = this grocery store is cheap), you’d use رخيصة.

In Arabic, an adjective is definite (الرخيص) only if the whole noun phrase is meant as “the cheap store” (a descriptive label).
Here, المتجر is definite because it’s identified by هذا (this makes the noun phrase definite), and رخيص functions as the predicate (“is cheap”), which is typically indefinite in nominal sentences. So:

• هذا المتجر رخيص = This store is cheap.
• هذا المتجر الرخيص (as a unit) = This cheap store (e.g., “This cheap store is… ”).

لكنني is لكن + ني:

• لكن = but
• ـني = me / I (object/pronominal ending used after certain particles; here it yields but I…)
  So لكنني لا أذهب = but I don’t go…

Both occur in writing: لكنني and لكني.
In careful MSA, لكنني is very common/standard (showing the full form before suffixes). لكني is also widely accepted and frequently used. Meaning doesn’t change.

The subject أنا (I) is optional because the verb أذهب already shows first person singular (the أ- prefix).
So لا أذهب is a complete sentence: I don’t go. Adding أنا can add emphasis/contrast: لكنني أنا لا أذهب إليه كل يوم = but I (personally) don’t go to it every day.

إلى means to. When you attach a pronoun to it, it becomes one word:

• إلى + ـه = إليه = to it / to him
  Here it refers back to المتجر (the store), so إليه = to it.

Yes. Arabic pronouns follow grammatical gender, not “natural gender.” Since متجر is grammatically masculine, you refer to it with ـه (often glossed as him/it): إليه = to it.

كل يوم (every day) is an adverbial time expression and is often placed at the end, but it can move for emphasis:

• لكنني لا أذهب إليه كل يوم (neutral)
• لكنني كل يوم لا أذهب إليه (possible but marked/awkward in many contexts)
• لكنني لا أذهب إليه يوميًا (using يوميًا = daily)

In fully vocalized MSA, يوم after كل is often genitive: كلَّ يومٍ (because كل is in an iḍāfa-like construction).
In normal unvowelled writing, you usually just see كل يوم. Case endings are typically omitted in everyday texts.

لكن can connect many types of clauses. Here it contrasts the first idea (it’s cheap) with a different action/habit (but I don’t go to it every day). Arabic doesn’t require the two sides to be the same sentence type.

Yes, in fully vowelled text you might see رخيصٌ (indefinite nominative ending) because it’s the predicate of a nominal sentence. It still means (is) cheap. In most real-world writing, the final vowels aren’t shown.

Not in this specific structure. رخيصًا (accusative) would be used in certain other patterns (e.g., after some verbs like وجدتُ المتجرَ رخيصًا = I found the store cheap).
Here, رخيص is a predicate of a nominal sentence and is (in full case marking) nominative: رخيصٌ.

You can, but it changes the structure slightly:

• لا أذهب إليه = I don’t go to it (explicit “to it”)
• لا أذهب هناك = I don’t go there (using there)
  Both can work depending on what you want to emphasize (the destination as “it” vs. “there”).