Ben süpürge alıyorum.

Turkish does not have definite or indefinite articles equivalent to English “a” or “the.” A noun without any suffix generally means “a/an” or “some.” If you want to emphasize “one” of something you can use bir, but it’s optional:

• Ben bir süpürge alıyorum = “I’m buying one vacuum.”

In Turkish, only definite direct objects take the accusative suffix -(n)i.

• Indefinite object (something non‐specific): süpürge alıyorum (“I’m buying a vacuum [not a particular one]”).
• Definite object (a specific vacuum): süpürgeyi alıyorum (“I’m buying the vacuum that we talked about”).

No. The personal ending -um on alıyor-um already shows that the subject is “I.” Native speakers often drop Ben unless they want to add emphasis or contrast:

• Süpürge alıyorum. (neutral)
• Ben süpürge alıyorum. (emphatic “I am the one buying…”)

It’s the present‐continuous form of almak (“to take/buy”), built in three steps:

1. al (stem)
2. • progressive suffix -(i)yor → alıyor (vowel harmony: a (back) → ı in -iyor)
3. • first‐person singular ending -um (vowel harmony: o (back, rounded) → u) → alıyorum

Use the question particle -mı/-mi/-mu/-mü (vowel‐harmonized) after the verb and the appropriate personal ending:

• Süpürge alıyor musun? = “Are you buying a vacuum?” (2nd person singular)
• Süpürge alıyor muyum? = “Am I buying a vacuum?” (1st person singular—rarely used)

Turkish is typically Subject-Object-Verb (S-O-V):

• Ben (Subject)
• süpürge (Object)
• alıyorum (Verb)
  You can move elements around for emphasis, but the verb usually comes last.