Translation:I ask Mr. Tanaka to close the door.
Technically speaking though, さん doesn't directly translate into Mr/Mrs/Ms; you could use it in scenarios where Western titles like that wouldn't make much sense. So I think that just "Tanaka" should be accepted in this instance.
Also, reported it, but for some reason even though it suggests Mr/Mrs/Ms on hover, Duo only accepts "Mr" right now for this specific question...
Imagine a situation where this sentence could be used, for example, it's a meeting. Tanaka is late. He enters the room and leaves the door open. In this situation, if address Tanaka, you are likely to say: 田中さん、ドアを閉めてください。 But before you would say that, you would probably think first: 田中さんに"ドアを閉めて"と頼みます。 - I'll ask Tanaka to close the door.
No, I realize that there are people like you who think so, and I fall somewhere in the middle. I mean yes will is used for predictions, and is therefore usually technically incorrect for the intended first-person singular meaning, sounding irresponsible and such, but there are also exceptions. Hence I recommend to use I'll instead of I will or I shall. Too many ignorant people on both sides . . .
They do, or at least I think so. At least some of the time, even in proper English. Plus there are so many people who use only I will incorrectly like this site. It's a regular Americanism. Conversation teachers seem to go rather with I am going to, which is supposedly more common, as it is on this site also.
But I'll is a contraction and not formal English, and is also short for I shall. Plus it's just so much shorter.
Everything before と (well, between 田中さん に and と if you want to be REALLY exact) is a subordinate clause - it's reported speech. We didn't personally witness the speaker telling 田中さん to shut the door - the speaker is telling us that s/he is asking or will ask 田中さん to shut the door. That is why it is とand not を. I asked Mr/Ms Tanaka to "reported speech/speaker's account of what happened". Thoughts are also conveyed in this way - 何々 と思います where the information in front of とis the speaker's thoughts.