What are some (bioinformatic) methods to characterize potentially novel gene transcripts?
It sounds like you have considered most of the obvious alternatives (and thank you for clarifying the question).  I suppose the first question in an alternatively spliced transcript with a retained intron is whether the open reading frame of the protein is maintained.  If there is a termination codon that now becomes in-frame due to the intron then the protein would ordinarily be truncated.  The mRNA could also get targeted for degradation due to the nonsense-mediated decay pathway (i.e., from introducing an early stop-codon).

If the alternatively-spliced transcript lacks any significant ORFs, or if there is an ORF but no suitable translation initiation codon then you have entered the realm of non-coding RNAs, of which there are two loose categories, microRNAs (there are several classes) or long non-coding RNAs.  most miRNAs have recognizable sequence motifs, may be contained in mir-Base, and may be complementary (in part) to a regulatory target elsewhere in the genome. lncRNAs are less well-defined. Perhaps the best criteria are transcripts that are stable enough to be detected, but lack noticeable protein coding features.  I don't even think there is a consensus on the minimum length of a transcript to be considered an lncRNA.

What sort of databases have you searched? For example the ancient dbEST contains all kinds of short cDNA reads from all over the planet, dating back to 1992 (or so).  Evolutionary conservation of exons and splice-sites, or reading frames can all be used to support the hypothesis that a transcribed region has a biological function.

For a really deep fishing expedition you can use tblastn, etc. to take a potential translated ORF and search all 6 conceptually translated reading frames of all the sequences in a database--it takes longer, and there can be many spurious matches that you have to search through, but if you are studying something that has never been annotated it might be worth considering.

RE: folding RNA (secondary structure) every sequence can be folded into some kind of structure (try it and see), but structurally conserved compact loops typically fall into one of a small class of selected families (like GNRY). Stems are easy to find, but if the single-stranded loop is huge, how likely is that structure to fold up in real time?

RE: TFBS not clear to me at all what you mean by this. Transcription factors bind dsDNA not ssRNA. There are families of protein domains that bind RNA but not typically with such sequence-specificity.  So unless this alternatively spliced transcript is being transcribed from some type of alternative promoter, I am not sure where you would even be looking for TFBS.  It's true that some large introns can indeed contain regulatory regions for the gene as a whole (cf. immunoglobulin heavy chain enhancer is conveniently stored in an intron), but it is not clear to me if that would have an affect when transcribed into RNA.

I do like your idea that a retained intron could now contain a binding site for a regulatory miRNA, but if I understand the theory this would lead to the transcript being degraded--so it would be more difficult to detect.