When using anything involving a BEC that the voltage regulator almost always works by losing some voltage, some more than others. When the supply voltage drops too low, different ones react differently. Usually the output voltage drops, sometimes, and it might depend on the load, it cuts off. Some radios can tolerate a bit of voltage drop, some don't and switch themselves off until they get the required voltage back. There are BECs out there that are able to step up from a lower voltage as well as step down, but they are "exotic" and therefore normally expensive, and don't get used in low cost products, or products like RC gear where price is important.
Using a separate battery gets round marginal voltage supply problems as long as the red wire from the ESC to radio is disconnected. A BEC trying to charge a battery normally does neither the battery nor the BEC any favours, some BECs do not respond well to having a higher voltage battery connected to their output.
A 7 Amp motor will only draw 7 Amps under the right load conditions. A resistance controller needs to be well matched to its motor under the assumed load. An ESC is less particular. What will drive a lot will just as well drive a little. While the 7 Amp motor might only rarely actually draw 7 A, it is usual practice to double that figure and pick the next higher rated ESC. So 15A is a good start, but any higher rating will work as well, with the exception of the "320A" ESCs which are widely pushed on t'interweb.