2HDMs with non-standard flavor structures are a powerful tool that can be utilized to address many open problems in physics. In analogy the four, well studied, models with natural flavor conservation (type 1, type 2, lepton-specific, and flipped), we examine four setups that implement an approximate U(2)5 flavor symmetry acting on the first two generations which provides the SM-like, heavy scalar, and charged Higgs bosons with interesting and non-standard phenomenology. We identify interesting and sensitive search channels for models where we generate the CKM in the down sector and up sector. We study the effects on low energy flavor violating processes such as Bd, Bs, and D meson mixing and from rare decays B decays. We further augment this framework by a real scalar singlet that, protected by a Z2 symmetry, provides a particle Dark Matter candidate. We show that this setup allows for doubly blind spots at both collider searches and direct detection experiments. Opening up large portions of parameter space that are ruled out in simpler Dark Matter models. Finally, we explore how an additional source of mass generation with a non-standard flavor structure can be utilized in order to reconcile the twin Higgs model with cosmology. We explore the possibility of striking displaced decay signatures that can occur at the LHC due to the production heavy Higgs.