watasi ha kagi wo tukue no ue ni okimasu.

は is the topic marker. It tells the listener “this sentence is about 私 (‘I’). In contrast, が is the subject marker that often introduces new or specific information.

• With 私 は …, you’re saying “As for me …”
• With 私 が …, you’d be emphasizing “I (and not someone else) …”

In everyday speech, は is much more common when you simply want to say “I do X.”

の is the genitive (possessive) particle, linking two nouns. Here 机 の 上 literally means “the top/above of the desk,” i.e. “on top of the desk.”

• A の B = “B of A” or “A’s B”

• に marks a destination or goal: you’re putting something to that location.
• で marks where an action takes place (e.g. “I study at school” = 学校 で 勉強します).

Since 置きます is an action of placing something onto a spot, you use に.

Yes. Japanese word order is flexible as long as the verb stays last:

• 私 は 鍵 を 机 の 上 に 置きます
• 鍵 を 机 の 上 に （私 は）置きます

• 机 の 上 に 鍵 を 置きます

  You can also drop 私 は if context makes “I” clear:

• 鍵を机の上に置きます。

The -ます form is the polite form used in formal or unfamiliar situations.

• Polite present/future: 置きます (okimasu)
• Dictionary (plain) form: 置く (oku)

In casual speech you’d say 鍵を机の上に置く。

Japanese non-past covers both present and future.

• 鍵を置きます。 can mean “I place the key” (habitual) or “I will place the key” (planned).
  Context usually tells you which one.