Why aren't there barlines in the piano solo parts in Beethoven's Piano Concerto No. 5?

This is a cadenza. The lack of barlines is, as you suppose, because Beethoven intends the performer to play in a rhythmically free, quasi-improvised manner. The pitch durations are intended to be suggestive -- relative to each other -- but not necessarily exact.

Another example is the opening of Chopin's Etude in C# minor, op. 25 no. 7. Notice that there is no time signature at the beginning. However, the editor of the score in the video has put in some suggested barlines as a guide to the performer. On the other hand, on IMSLP, Mikuli's edition (below) does not include those.

Another example comes from the final bar of Chopin's Etude in E minor, op. 25, no. 7 (Mikuli's edition again)

Also, regarding your question "How can '16th' note have any meaning without barlines", the answer is that the note values are not influenced by the presence of barlines. In strict tempo, a 16th is always the duration of two 32nds or half of an 8th. The note durations are only relative to each other. Barlines are, essentially, a convenience for reading -- so that you can easily see groupings of beats.