Heterogeneous Internet of Things (IoT) networks, which operate using various protocols and spectrum bands like WiFi, Bluetooth, Zigbee, and LoRa, bring many opportunities to collaborate and achieve timely data collection. However, several challenges must be addressed due to heterogeneous data patterns, coverage, spectrum bands, and mobility. This paper introduces a cross-technology IoT network architecture design that facilitates collaboration between service providers (SPs) to share their spectrum bands and offload computing tasks from heterogeneous IoT devices using multi-protocol mobile gateways (M-MGs). The objective is to minimize the age of information (AoI) and energy consumption by jointly optimizing collaboration between M-MGs and SPs for bandwidth allocation, relaying, and cross-technology data scheduling. A pricing mechanism is presented to incentivize different levels of collaboration and matching between M-MGs and SPs. Given the uncertainty due to mobility and task requests, we design a cross-technology federated matching algorithm (CT-Fed-Match) based on a multi-agent actor-critic approach in which M-MGs and SPs learn their strategies in a distributed manner. Furthermore, we incorporate federated learning to enhance the convergence of the learning process. The numerical results demonstrate that our CT-Fed-Match-RC algorithm with cross-technology and relaying collaboration reduces the AoI by 30 times and collects 8 times more packets than existing approaches.