Do transcripts always start and end with exons?
Most (almost all, AFAIK) mRNAs and lncRNAs start with exons for the reasons already mentioned by David. In a typical splicing event, the nucleotide that is 5' to the splice donor site (lets call it pre-donor) and the one that is 3' to the acceptor site (lets call it post acceptor) are joined together and the intronic sequence between them is removed. 

If you look carefully at the mechanism, you'll notice that the pre-donor (i.e. the 3' end of the first exon) renders a nucleophilic attack on the phosphorous of the post acceptor (i.e. the 5' end of the second exon) which leads to the release of the intron.


  
   Image courtesy: http://mips.helmholtz-muenchen.de/proj/yeast/reviews/intron/spliceo_splicing.html 


Therefore there is always an exon preceding an intron, in the transcript.

However, RNA processing can occur via a different mechanism in which the ends are just chopped off. This happens in tRNA processing, where RNAseP chops off a portion 5' to the mature tRNA region, called the leader sequence (see the figure below). I am not aware of such a mechanism happening in the case of mRNAs but a possibility certainly exists. One possible reason for mRNAs not having this kind of mechanism is because their 5' end is capped (co-transcriptionally) and cleavage of the 5' would lead to loss of cap and destabilization of the mRNA (all guesses, though). Similarly, the 3' of mRNAs is polyadenylated which also renders stability to them. Many lncRNAs are also capped and polyadenylated but there are exceptions (mRNAs too). 

Since this mechanism is not very common, there is no systematic name for the excised ends. For the case of tRNA, it is simply called the "leader" sequence.


  
   Image courtesy: Leigh &amp; Lang, 2004