"I put the wallet in the bag."
Would it sounds strange/wrong to a native speaker if I used "さいふをカバンに" instead of "カバンにさいふを"? I'm not sure I fully understand the proper structure for sentences with both an indirect and direct object marker yet...
Don't worry about it, both are fine and will be understood. You can pick up the "normal" way fast enough if you use the language in practice, until then either will do.
It might sound a little odd, but either should be perfectly intelligible.
No, に marks location and 入れます is a transitive verb (someone did something TO something to go into something). を is needed to mark the object going into and に is needed to mark what it went into.
That can be used but を would be a more grammatically correct option here.
Please correct me if I'm wrong.
I answered this in past tense and got it wrong. Like the present tense of "put" and past aren't the same. Reported.