Is it possible to have multiple stop codons in one exon?
An exon can have multiple stop codons but the first codon will terminate the ORF. The remainder of the exon will be a part of the 3'UTR. 

However, there are some cases in which stop codon readthrough happens [1]. In these cases an internal stop codon does not terminate the translation and the ribosome reads through it. During this process an aminoacyl-tRNA successfully competes with the release factor. This is quite evident in the case of selenocysteine and pyrollysine- tRNAs which bind to the UAG (amber) stop codon in certain organisms [2,3] (this is also referred to as amber suppression). The exact mechanisms that affect stop codon readthrough are not elucidated but features in the mRNAs are likely to play a role in this. In case of amber suppression, the relative rarity of the UAG stop codon and lower concentrations of release factor-1, can be major factors.


I don't know what you mean by "beginning of the exon" but there are short ORFs (&lt; 100 codons) which can code for small peptides [4]. Small peptide as small as 6 amino acid residues has been reported but most of the known small peptides are ~30-100 residues long. 



References:
[1] Loughran, Gary, et al. "Evidence of efficient stop codon readthrough in four mammalian genes." Nucleic acids research 42.14 (2014): 8928-8938. 
[2] Agafonov, Dmitry E., et al. "Efficient suppression of the amber codon in E. coli in vitro translation system." FEBS letters 579.10 (2005): 2156-2160. 
[3] Wikipedia: Expanded genetic code 
[4] Andrews, Shea J., and Joseph A. Rothnagel. "Emerging evidence for functional peptides encoded by short open reading frames." Nature Reviews Genetics 15.3 (2014): 193-204.