If something is publicly visible in the Internet, does it mean I can use it for research?
There are at least three largely-unrelated questions buried in here. Disclaimer: I am not a lawyer, this is not legal advice, and in any case its correctness may vary a little according to where you are.

The first thing to establish is that "appearing on the internet", whether in Google Images or elsewhere, does not mean that something is in the public domain. Something may be public domain, or it may be copyrighted with a permissive license, or (most commonly) neither of these may apply - although you can configure Google Images to search for pictures with public domain and/or Creative Commons licenses.

Now, the three questions:


If something is available online can you use it in your research? I don't see any copyright reason why not, since you're not distributing it; downloading an image to perform research is not fundamentally any different to downloading an image (in your browser) to view it. 
You might need to be careful that your research (which you presumably do want to distribute) is not a "derivative work" of the copyrighted material. Reporting something like "50% of pictures on the internet are of cats" would not be a problem, but for less clear-cut cases - for example where part of the original material remains - you might want to check.
If something is available online can you redistribute it? No, or at least not by default. Not unless it is public domain or licensed in such a way as to permit this. "Fair use" considerations may modify this answer in some jurisdictions.
Can you distribute a list of the URLs of copyrighted works that others could download for their own research? Probably, but there have been some attempts at really bizzare laws in this area in some countries, due to websites that provide links to pirated movies, etc., that they do not host.