This paper completes in a nice way the research about weighted inductive limits of spaces of continuous functions with values in a Fréchet space. These spaces are defined as follows: let \(V=(v_n)_n\) be a decreasing sequence of strictly positive continuous functions on a completely regular Hausdorff topological space \(X\) and let \(E\) be a non-normable Fréchet space. The weighted (LF)-space \(VC(X,E)\) is defined as the countable inductive limit \(\text{ind}_n Cv_n (X,E)\) of Fréchet spaces of continuous functions. Its projective hull is denoted by \(C\overline{V}(X,E)\). The investigation of these weighted (LF)-spaces of vector valued continuous functions was initiated by \textit{K. D. Bierstedt} and the reviewer [NATO ASI Ser., Ser. C 287, 195--221 (1989; Zbl 0708.46037)]. Several problems remained open. This research was continued by \textit{A. Galbis} [J. Aust. Math. Soc., Ser. A 50, No. 2, 233--242 (1991; Zbl 0746.46027)]. Using deep results due to Vogt and Wengenroth, \textit{E. M. Mangino} [Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 90, No. 3, 171--173 (1996; Zbl 0886.46002)] was able to solve some of the open problems in 1996 in the case of sequence spaces instead of spaces of continuous functions. Extending previous work by \textit{E. Mangino, S. Dierolf, L. Frerick} and \textit{J. Wengenroth} [in: Functional analysis. Proceedings of the first international workshop held at Trier University, Germany, September 26--October 1, 1994 (W.~de Gruyter, Berlin), 293--304 (1996; Zbl 0908.46002)], the author proves that the (LF)-space is regular if and only if it satisfies condition (M) of Retakh, and this is in turn equivalent to a condition on \(V\) and \(E\) introduced by Vogt. As a consequence, if \(X\) is locally compact and \(VC(X,E)=C\overline{V}(X,E)\) holds algebraically, then \(VC(X,E)\) is complete, thus solving an open problem by Bierstedt and the reviewer. Extending a result she had proved in [Proc. Am. Math. Soc. 128, No. 2, 583--588 (2000; Zbl 0948.46003)], the author also shows the following nice result: the (LB)-space \(VC(X)\) is boundedly retractive if and only if every compact subset of \(VC(X)\) is contained in the closed absolutely convex hull of a null sequence, a result which should be compared with a consequence of the Banach-Dieudonné theorem for metrizable spaces. In the last section, a result of Mangino for vector valued sequence spaces about the topological and algebraic identity \(VC(X,E)=C\overline{V}(X,E)\) is obtained. It is characterized in terms of a different condition on \(V\) and \(E\) which was also introduced by Vogt.