Is it just the て-form of ます? So, could I just use 食べまして instead of 食べて? Leading to my next question (everything from here on is assumed):
Since 食べまして is the polite form of 食べて, why wouldn't you use 食べましてください instead of 食べてください, isn't that even more polite? But then again, I never heard of 食べましてください, so what is it about the ～まして part?
Yes, it is the -て form of ます. But it's a little more restricted, so you need to be a bit careful.
To be polite, you normally only need to use the です/ます form for the final verb. Any other verbs can be in their normal -て form. But if you really want to be polite, then you can put the other verbs in their polite -ます form, obviously resulting in -まして. It is often found in keigo sentences.
ご迷惑をおかけしまして申し訳ありません。
However, you cannot combine it with ください. The phrase 「食べましてください」 is highly unnatural, if not ungrammatical. To give a request, stick with the regular -て + ください (or use a keigo expression if you still want to be very polite).
That said, sentences can end with -まして, but it is never a request (as said before, this isn't typically used for requests); it is a cut-off mid-sentence, with the second part unsaid.
田中さんからお電話がありまして - Tanaka-san called...
