Bluetooth Low Energy general questions
How does the Peripheral receives the request. Is the antenna in receiving mode, while sending advertisements? If so, is this receiving mode always on, reacting to any connection request (even without sending advertisements)?

Advertising, scanning and establishing connection all happens in the Link Layer. There are three advertisement channels in BLE. The advertiser will send advertisements on all three channels. The scanner will check these channels for advertisements and reply with a connection request if needed. The advertiser is sending advertisements and checking for connection requests from the scanner at the same time.

Depending on the GAP role (advertiser, scanner) the device will react to advertisements/connection requests.

Something about your questions:

Third: As soon as connection is lost unintentionally both roles will go in disconnect state. It depends on your application what should happen after that (reconnection?). When a connection is properly terminated (by sending a disconnect message), the connection is lost. One advantage of BLE is that a connection can be re-established very fast from scratch because it requires fewer messages to establish a connection (compared to BT Classic).

Fourth: It is possible to send some data with the advertisement message or send data as a response to a scan request.

Sixth: Advertisements contain an advertiser address and other information about the device which is sending the advertisements (name, address, which profiles are enabled). With this information you can determine if a device is interesting for you.