Can this randomized greedy algorithm be made online? Or being proved impossible?
as far as I can tell after looking into this somewhat, there is a misunderstanding of the paper in the question. the question states that some of the algorithms are offline vs some online. but the algorithms in the paper are all online it appears (but agree the authors are not clear on this point). that is what is accomplished by having random edge orderings-- the random edge ordering reflects edges coming "online" in whatever order, right? 

the authors seem to make no reference to "offline" algorithms in the paper and they refer to "online" only in the introduction that I see. they dont state this but the 1st algorithm seems to allow edges to come "online" in "batches" (ie random reordering in stages) whereas the 2nd algorithm does not.

while you dont state it exactly you appear to be checking the math in proof for theorem 2.3 in the paper (p3) which establishes a lower bound on the number of colors in a tree and the authors assert it works for both algorithm 1 and 2 ("for the 2nd algorithm.. the same bound holds"). the proof is that no less than $1.23\Delta$ colors ($\Delta$=graph degree) are sufficient for all trees. it seems if this theorem is wrong as you are suggesting then one should be able to exhibit a tree that can be colored with fewer than $1.23\Delta$ colors using those algorithms. from what I can tell your math is off because you are not expressing it using the same formulas in the proof eg you do not calculate the exponential formula in the proof.

a way to empirically verify this lower bound would be to generate many random trees, or some sample from some distribution, run the two algorithms and verify that they always use at least $1.23\Delta$ colors.