Symbols of animals
Unfortunately, Detexify doesn't find anything, and neither does The Noun Project that Martin Scharrer wrapped into a .sty file.
However, if you manage to find a suitable symbol (with the right license), you could just define your own command to include it in a document. For example, on her site mytimetocraft.com, Beth has published a nice SVG of a dolphin. The images posted on that site are free for commercial and non-commercial use.
You could either convert that SVG to a PDF file, or you could just extract the path from the file and let TikZ draw it for you. By setting the scale factor as a product of the ex unit, the symbol will scale with the surrounding text.
Look, a \Dolphin!
\Huge Wow! It's so big and blue! \Dolphin[blue,xscale=-1]
will yield
Here's the code for the above example:
\documentclass[a4paper,11pt]{article}
\usepackage{tikz}
\usetikzlibrary{svg.path}
\newcommand{\Dolphin}[1][]{%
\tikz \fill [scale=1ex/500,yscale=-1,#1] svg "M 240,792 C 236.7964,782.29301 227.32595,769.18429 218.95458,762.8695 C 210.58318,756.55478 193.07658,741.91749 180.05096,730.34232 L 156.36806,709.29648 L 141.17601,721.53827 C 132.82037,728.27131 116.86485,736.20204 105.71929,739.16215 C 94.57372,742.12228 76.825609,750.15889 66.279001,757.02126 C 42.749986,772.33104 35.286663,768.91329 39.793018,744.89235 C 49.786747,691.62137 86.56609,637.47901 123.0161,622.38091 C 133.32885,618.10921 138.97291,610.6592 143.54681,595.28094 C 156.31386,552.35575 185.13223,485.10659 210.50659,439.02657 C 285.62548,302.61033 377.12604,221.91941 502.90248,181.17368 L 545.4538,167.38902 L 523.9559,153.16241 C 512.13201,145.33777 493.01385,137.67119 481.47106,136.12559 C 441.72739,130.80384 454.06939,100.38528 495.97229,100.38528 C 525.1625,100.38528 588.32595,117.28333 612.47161,131.55225 C 631.20187,142.62093 638.05903,143.46536 711.12662,143.70143 C 826.20118,144.07318 875.39668,159.99674 936.18511,216.548 C 957.70508,236.568 977.32088,249.47994 997.76359,257.08148 C 1037.1384,271.7229 1050.5601,280.83354 1050.5601,292.92001 C 1050.5601,304.96929 1043.1872,306.72156 979.27795,309.86153 C 937.41314,311.91844 922.54209,315.36478 870.35529,335.10414 C 837.1142,347.67736 806.21965,359.09969 801.70072,360.48707 C 797.18181,361.87448 788.5903,373.11962 782.60856,385.47631 C 769.33352,412.89887 732.8923,450.74636 708.71893,462.21739 C 687.47223,472.29962 645.39097,477.58347 636.22902,471.31951 C 626.59388,464.73198 633.77941,451.7414 651.00501,444.60631 C 668.76327,437.2506 688.54805,412.99772 693.73219,392.22992 L 697.39133,377.57125 L 659.92998,381.16743 C 639.32621,383.14537 594.48504,391.40641 560.28293,399.5254 C 408.28252,435.60753 289.59011,503.28314 223.04114,591.81285 L 199.83212,622.68772 L 222.77838,652.2359 C 249.05732,686.07569 265.30365,731.48301 265.30365,771.09095 C 265.30365,810.05728 250.10447,822.61677 240,792 M 885.93164,242.20228 C 889.77822,235.9783 875.83562,229.90601 868.33788,234.5399 C 865.37262,236.37259 864.25805,239.99423 865.8612,242.58811 C 869.72788,248.8446 881.97191,248.60916 885.93164,242.20228 z";%
}
\begin{document}
Look, a \Dolphin!
\Huge Wow! It's so big and blue! \Dolphin[blue,xscale=-1]
\end{document}
If you're interested in what symbols you can access through LaTeX commands, you might find the Comprehensive LaTeX Symbol List to be of some use to you.
But you won't find many animal symbols there. The few that I have managed to glean are not actually dingbats but symbols from the Phaistos Disk, accessible through the package phaistos.
Examples from the symbol list I've mentioned (p. 81):
(The cat symbol is lovely, by the way.) Still, you won't find any dolphins there, I'm sorry to say!
There are dedicated packages to add animals (and other beings) to latex:
\documentclass{standalone}
\usepackage{tikzlings}
\usepackage{tikzducks}
\begin{document}
\tikz\bear;
\tikz\penguin;
\tikz\marmot;
\tikz\koala;
\tikz\owl;
\tikz\coati;
\tikz\snowman;
\tikz\rhino;
\tikz\moles;
\tikz\sloth;
\tikz\pig;
\tikz\cat;
\tikz\hippo;
\tikz\panda;
\tikz\mouse;
\tikz\duck;
\end{document}
(the code above need the development version of the tikzlings package, which is available from https://github.com/samcarter/tikzlings)