Jesper Grodal's reference to Bourbaki is a reasonable one for these questions, including 1). There are also two volumes in the Springer GTM series which treat many aspects of compact Lie groups, including the book by Bump indicated and the earlier text GTM 98 (1985) by Brocker & Tom Dieck on Representations of Compact Lie Groups where V.7 treats the fundamental group and related matters thoroughly. Naturally the notation differs somewhat in such sources, but the answers to the questions raised here are all standard and arise from early work of Cartan, Weyl, and others. In general, the topology of a semisimple Lie group depends just on the topology of a maximal compact subgroup.
In the setting of abstract root systems, motivated by the theory of semisimple Lie groups over the complex field and their Lie algebras or by compact semisimple Lie groups, the notion of "fundamental group" focuses on the quotient of the abstract weight lattice by the abstract root lattice. In semisimple groups or Lie algebras, the actual weight lattice (or character group) of a maximal torus can vary from the root lattice to the full abstract weight lattice, but the quotients in any case are finite and easily computable for each simple type. Moreover, the abstract fundamental group for a given root system is realized internally as the center of a simply connected group.