How to create an Access Point from the current WiFi connection?

can an adapter have both a hotspot and be connected to a WiFi network at the same time?

Yes , it's possible to create a hotspot wifi from the same wifi connection

How do I share the Internet of the currently connected network over a hotspot in Linux?

You should use create_ap

Install the dependencies :

sudo apt-get install dnsmasq haveged hostapd build-essential

To install create_ap run:

git clone https://github.com/oblique/create_ap
cd create_ap
sudo make install

enable/start the service:

service create_ap start

To check the status run:

service create_ap status

Using the systemd features (it is not available on Ubuntu 14.04):

sudo systemctl start create_ap.service
sudo systemctl enable create_ap.service

Create the AP

To create your access point run:

sudo create_ap wlan0 wlan0 MyAccessPoint MyPassPhrase

Update for Ubuntu 14.04:

the hostapd version in Ubuntu 14.04 is buggy and doesn't work properly. To get AP-Hotspot to work with hostapd in Ubuntu 14.04, you need to downgrade hostapd and use apt to hold the package so it's not upgraded.

Remove the current version of hostapd:

sudo apt-get remove hostapd

64bit

cd /tmp
wget http://old-releases.ubuntu.com/ubuntu/pool/universe/w/wpa/hostapd_1.0-3ubuntu2.1_amd64.deb
sudo dpkg -i hostapd*.deb
sudo apt-mark hold hostapd

32bit

cd /tmp
wget http://old-releases.ubuntu.com/ubuntu/pool/universe/w/wpa/hostapd_1.0-3ubuntu2.1_i386.deb
sudo dpkg -i hostapd*.deb
sudo apt-mark hold hostapd