False - a repeater is a simple layer 1 device. It would not have a need to operate at the network layer.
Network layer
A repeater. Repeaters work only on layer 1 (Physical) of the OSI model.
Physical layer = Hub, NIC, Repeater. Data Link Layer = Switch, Bridge. Network Layer = Router.
G repeater will not work with N network, unless your router is configured to work in compatibility mode. It means that your router can work with both G and N networks. Thus when you connect your G repeater it will extend only your G network.
An active hub can act as a repeater hub. It may also work as an Ethernet hub, network hub, and a multiport repeater.
It is work at Network layer.
A network bridge joins two networks to make one. A network repeater extends one network to make it larger/more powerful.
Physical layer devices would be a hub or a repeater to connect multiple Ethernet segments together.
Repeater.
There is a good descrition of the network layer and TCP in the related link. NO. In general TCP and UDP are transport protocols that operates in Layer 4 Transport Layer
Very simple - most people who require a wireless repeater are those who need to extend their wireless network. This means a repeater would take the original signal from the network and repeat it, which extends this network. This is one of the ways which people are using to get around putting wires around their home or office.