Saya menaruh tas di bawah meja supaya tidak mengganggu.

Indonesian often turns a root word into an active verb by adding the meN- prefix.

• Root: taruh = (to) put/place (informal base)
• Active verb: menaruh = to put/place (something)
  The prefix changes form depending on the first sound of the root. With t, the t usually drops: meN- + taruh → menaruh.

They’re very close in meaning: both mean to put/place. Common nuance:

• menaruh: more everyday, neutral, often used in spoken Indonesian.
• meletakkan: slightly more formal/explicit about placing an object down. In most situations like this sentence, either would sound fine.

You usually include an object because menaruh is transitive (it normally takes an object). Here the object is tas (bag), so it’s complete.
You can specify ownership for clarity:

• Saya menaruh tas saya di bawah meja… = I put my bag under the table…

di bawah meja is a location phrase:

• di = in/at (location marker)
• bawah = under/below
• meja = table
  Indonesian doesn’t have articles like a/the, so meja can mean a table or the table depending on context.

Standard Indonesian writes it as two words: di bawah.

• di as a location marker is written separately.
  dibawah is commonly seen informally, but it’s considered non-standard.

All three can mean so that / in order that, but with slightly different tone:

• supaya: very common, neutral.
• agar: a bit more formal/written.
• biar: more casual; can also feel like let it… / just so… in some contexts.
  Your sentence could also be:
• …agar tidak mengganggu. (more formal)
• …biar tidak mengganggu. (more casual)

Indonesian often uses supaya/agar + clause (a full verb phrase) to show purpose. So supaya tidak mengganggu is a natural way to express so that it doesn’t disturb.
Using untuk is possible but tends to sound different:

• untuk is more like to / for and often pairs with nouns or more “goal-like” phrasing.
• supaya/agar is the most straightforward for “so that + verb”.

It’s typically understood as the bag (or the situation) not I:

• …so that it (the bag) doesn’t disturb (anyone).
  Indonesian often leaves the subject implicit when it’s obvious from context.

mengganggu can be used:

• transitively: mengganggu orang = disturb someone
• intransitively/implicitly: tidak mengganggu = not be a nuisance / not get in the way
  In this sentence, the affected person(s) is left unsaid because it’s general: not to bother anyone / not to be in the way.

No.

• tidak negates verbs and adjectives: tidak mengganggu = not disturbing
• bukan negates nouns/identity: bukan dokter = not a doctor
  Since mengganggu is a verb, tidak is correct.

Yes. Indonesian allows that reordering, especially in speech or writing for emphasis:

• Saya menaruh tas di bawah meja supaya tidak mengganggu.
• Supaya tidak mengganggu, saya menaruh tas di bawah meja.
  Both are natural; the second one highlights the purpose first.

It’s neutral and very natural in everyday conversation.
To make it more formal, you might choose slightly more formal words:

• Saya meletakkan tas di bawah meja agar tidak mengganggu.