The answer to your question in the last paragraph is "Yes", it is a contents of the well-known Sobczyk Theorem, see Lindenstrauss-Tzafriri, Classical Banach spaces, vol. I.

The answer to the first question is still negative. You can find an example of this type in W.B. Johnson, J. Lindenstrauss, Examples of L1 spaces, Ark. Mat. 18 (1980), no. 1, 101–106.

The answer is positive if the space is reflexive - you can consider the weak limit of the projections.

**Added on January 5, 2018.** Many other examples giving a negative answer to the first question can be constructed combining the result of Zippin [Israel J. Math. 26 (1977), no. 3-4, 372–387] stating that any separable infinite-dimensional Banach space which is not isomorphic to $c_0$ can be embedded into a separable Banach space as an uncomplemented subspace, and the well-known fact that a Banach space can represented as a closure of the union of increasing sequence of finite-dimensional subspaces which are uniformly close to $\ell_\infty^n$ without being isomorphic to $c_0$ (see, e.g., very exotic examples in Bourgain, Pisier [Bol. Soc. Brasil. Mat. 14 (1983), no. 2, 109–123].)