المشكلة في الشركة اليوم، وأنا أنتظر حلا.

It’s a nominal sentence (جملة اسمية). Arabic often omits the verb to be in the present tense.
So المشكلة في الشركة اليوم is essentially: The problem (is) in/at the company today.
Grammatically:

• المشكلة = مبتدأ (topic/subject)
• في الشركة اليوم = خبر (predicate), a prepositional phrase + time adverb

Yes, you could, but the meaning shifts:

• المشكلة = the problem (a specific, known problem)
• مشكلة = a problem (unspecified/new information)

In this sentence, using المشكلة suggests both speaker and listener know which problem is meant.

في الشركة literally means in the company, but in context it can also mean at the company (i.e., at the workplace / within the company).
If you want to be more clearly “at the company (location),” you might also see في مقرّ الشركة (at the company headquarters) or في الشركة is often still fine depending on context.

Time words like اليوم commonly come after the place phrase, especially in neutral narration:

• في الشركة اليوم = at the company today

You could also move it for emphasis:

• اليوم المشكلة في الشركة = Today, the problem is at/in the company (more emphasis on “today”).

Correct: أنتظر already implies I. Adding وأنا is optional and usually adds emphasis or contrast, like:

• …and I (for my part) am waiting…

So:

• وأنا أنتظر… feels more expressive than just وأنتظر…, though both are grammatical.

أنتظر is the present tense (imperfect) verb, 1st person singular: I wait / I am waiting.
Root: ن ظ ر in the pattern انتظر (Form VIII), meaning to wait (for) or to expect.

In fully vowelled MSA it would often be written/spoken as أنتظرُ (with final -u in careful pronunciation).

Yes—orthographically in standard MSA, a solution as the object of أنتظر should be:

• حلاً (usually written without the shadda but with tanwīn fatḥ + the extra alif) And with full spelling including the doubled ل:
• حلًّا

What you see as حلا is basically حلاً but missing the tanwīn diacritic (and often missing the shadda). In normal unvowelled Arabic text, people may omit those marks, but the “extra” ا is still typical for tanwīn fatḥ.

It’s the direct object (مفعول به) of the verb أنتظر:

• أنتظرُ حلاً = I am waiting for a solution

Because it’s the object, it’s in the accusative (منصوب), which is why—when vowelled—it takes tanwīn fatḥ: حلاً.

Yes, and the meaning changes slightly:

• أنتظر حلاً = I’m waiting for a solution (any solution)
• أنتظر الحلّ = I’m waiting for the solution (a specific expected solution)

Both are correct; the choice depends on whether the solution is specific/known.

A careful, fully vowelled version could be: اَلْمُشْكِلَةُ فِي الشَّرِكَةِ الْيَوْمَ، وَأَنَا أَنْتَظِرُ حَلًّا.

Notes:

• المشكلةُ ends with -u (subject/topic in a nominal sentence)
• الشركةِ ends with -i after في
• اليومَ is often -a as a time adverb (ظرف زمان)
• أنتظرُ ends with -u in careful citation
• حلًّا is accusative with tanwīn fatḥ