Android - How to use an Android phone or tablet to make a projector presentation?
Update 2017
Much has changed since the writing of this answer. The old answer (see edit history) is no longer the best answer to this question.
I'd recommend getting a Chromecast and using it with a VGA adapter. This gives you the benefit of giving the presentation wirelessly, instead of leaving your phone tethered to a cable.
Most modern Android phones have this capability, but double check before you purchase if this is something important for you.
Make sure that the phone that you get has a micro-HDMI port, in addition to the traditional USB port. This will allow you to transfer video more easily.
As you noticed, there are two types of converters- the tiny ones, and the large bulky ones with a power supply. The difference is, as stated on the link you sent, that:
Unlike passive adapters Ligawo converter has a power supply, thereby allowing a built-in signal amplification and processing of the image. Provides the signal source sufficient current through HDMI This converter can also be used without power supply.
So yes, the larger ones with power supplies would probably be better in terms of signal quality, and could still be used without being plugged in. However, due to their bulk, the better signal processing (I'm not an expert on signal processing, so I don't know how large of a difference it would make- I've used the small kind, although with a laptop, and it worked perfectly fine) might not be worth it- it's up to you to decide which type is right for you.
As of how the resolution is scaled, it should work automatically for all, if not most, converters. Just double check if it says something like "Input resolution HDMI up to 1080p possible 480p - Output Resolution VGA" (from the link you sent). It shouldn't be a problem to downscale to a specific amount that you want- in the Settings -> HDMI -> HDMI Format, you should be able to change the output resolution. For more info, look here- it is two years old, but the features should still be the same with new phones running JB/KitKat.
As of the software to run, there are many free and premium solutions available in the Google Play store that will work well. If you prefer presenting with PDFs, PDF Reader or Office Suite 7 should work. However, PDFs are a somewhat unconventional presentation format (although they can work), so you should consider using actual presentation software for a better experience. Google Drive for Mobile would work well, since you could edit your presentation on a computer (or your phone) and then present it on your phone, plus it is free. You could also use Presentation Pal, or Office 2012: Presentations, or some other presentation software- there are lots out there, and there's bound to be one that works good for you. All presentation software in the play store should work well for your purpose.
I hope this helps!