<p dir="ltr">In this thesis, we develop an algebraic framework for colored, colored connected, semi-grouplike-flavored, and pathlike co-/bi-/Hopf algebras, which are essential in combinatorics, topology, number theory, and physics. Moreover, we introduce and explore simply colored comonoid, which generalises the notion of colored conilpotent coalgebra. The simply colored structure captures the essence of being connected and give unified treatment of all connected co-/bi-algebras. </p><p dir="ltr">As a consequence, we establish precise conditions for the invertibility of characters essential for renormalization in the Connes-Kreimer formulation, supported by examples from these fields. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the appearance of Brown style coactions. We also investigate the relation between pointed coalgebras and color conilpotent coalgebras. </p><p dir="ltr">Using these results, we interpret all relevant coalgebras through categorical constructions, linking the bialgebra structures to Feynman categories and applying our developed theory in this context. This comprehensive framework provides a robust foundation for future research in mathematical physics and algebra.</p>