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In this paper, a meshfree method using the deep neural network (DNN) approach is developed for solving two kinds of dynamic two-phase interface problems governed by different dynamic partial differential equations on either side of the stationary interface with the jump and high-contrast coefficients. The first type of two-phase interface problem to be studied is the fluid-fluid (two-phase flow) interface problem modeled by Navier-Stokes equations with high-contrast physical parameters across the interface. The second one belongs to fluid-structure interaction (FSI) problems modeled by Navier-Stokes equations on one side of the interface and the structural equation on the other side of the interface, both the fluid and the structure interact with each other via the kinematic- and the dynamic interface conditions across the interface. The DNN/meshfree method is respectively developed for the above two-phase interface problems by representing solutions of PDEs using the DNNs' structure and reformulating the dynamic interface problem as a least-squares minimization problem based upon a space-time sampling point set. Approximation error analyses are also carried out for each kind of interface problem, which reveals an intrinsic strategy about how to efficiently build a sampling-point training dataset to obtain a more accurate DNNs' approximation. In addition, compared with traditional discretization approaches, the proposed DNN/meshfree method and its error analysis technique can be smoothly extended to many other dynamic interface problems with fixed interfaces. Numerical experiments are conducted to illustrate the accuracies of the proposed DNN/meshfree method for the presented two-phase interface problems. Theoretical results are validated to some extent through three numerical examples.