Yes, you can read GATT DB entries from a connected BLE device and display those entries on a webserver you connect to over Wi-Fi that's running on the 4334X device.
The Smartbridge demo in the WICED SDK is the foundation for what you are trying to do.
Our partner, Inventek Systems has created a demo using their new demo board (which is a replica of ours, but running on a USB thumb drive) which takes our SmartBridge Demo a step further and allows devices like our WICED Sense to easily connect and display GATT DB values.
They have been showing this demo at the recent IoT seminars here in North America and it has been a big hit with customers: Broadcom IoT Seminars coming to a city near you! (Chicago and Minneapolis next week)
Take a look at the blog abuch put together here: Inventek Smart Bridge - ISM340-USB Demo - WICED Sense Kit
Can I read the iBeacon entries sent over BLE as well? A device sending the iBeacon message would not be connected to the base station. We need to add intelligence at the base station side to receive only certain messages but the BLE device will not be in the connected state.
You will want to take a look at the Smartbridge sample application and determine if it can be modified to meet your application requirements.