The Internet of Things and the next-generation networks have led to the generation, dissemination, and transformation of a huge amount of real-time information. The information is often governed by processes that evolve over time and/or space (e.g. on an underlying network). This thesis will develop theoretical foundations and algorithmic designs for time-critical decisions with real-time information extraction in networked systems and considers applications such as estimation and network coding in IoT and testing and isolation for COVID-19. In the first part, we study the timeliness of information transfer in communication networks. Timeliness was first captured and quantified through the metric of Age of Information (AoI) and has since become a new design criterion in communications. We provide an understanding of the topic when the multi-access and broadcast nature of wireless networks are also taken into account. Firstly, in random access channels, we propose decentralized selection and transmission policies to minimize the normalized time-average AoI, and state that selection and transmission are coupled and a separate treatment of the two may not be sufficient. Then, we consider the remote sampling and estimation problems in random access channels. The AoI is proved to be a good proxy of estimation error. Decentralized sampling and transmission policies are proposed to minimize the normalized time-average estimation error. Next, we extend the remote sampling and estimation problems to ad-hoc networks, where the theoretical analysis is intractable due to the increased dimensions of decisions and complex network topologies. We propose a graphical reinforcement learning framework with permutation equivalent properties, and we also prove the transferability of the proposed framework. We further shed light on the tradeoffs between timeliness and communication rate in broadcast networks. We find that coding is beneficial to the reduction of the AoI, and the benefits increases with the number of users. In addition, the tradeoff between the AoI and rate exists, and the system needs to sacrifice AoI to achieve a higher rate. Going beyond age minimization in wireless networks, in the second part of this thesis, we study the problem of testing and control of spread processes (e.g. infectious diseases such as COVID-19). This problem is another instance of time-critical decision-making with real-time information extraction. The spreading process is modeled not only evolves in time but also over an underlying network. We aim to design sequential testing and isolation policies to contain the spread as soon as possible (minimize the cumulative infections). We first formulate the problem as a minimization of a supermodular function. Then, we proposed an exploration-exploitation testing and isolation policy and a novel backward message-passing framework. We theoretically show that both backward updating and exploration processes are necessary for decision-making. Finally, simulations show our proposed policies outperform the state-of-the-art.