PNG-24 is superior to JPEG and PNG can contain transparency though it won't compete on filesize for photos. The PNG should be several times larger than a JPG (for most photos) anyway as it has to store all the colour spaces.
Hmm, a web browser has no concept of dpi thus it will appear different to say Photoshop (where you can see print and actual size, etc.) As far as the browser is likely to be concerned anything over 72 dpi is meaningless anyway (or for that matter so is 72 but that's another story).
In reality the dpi 'Text Size' is a "Logical Inch" computed value. Don't confuse "logical inches" with "real inches" - very different concepts.
For example we have three; 100px by 100px images, one at 72 dpi, 96 dpi and 300 dpi. It possibly will look the exact same size within a web browser but when printed it will differ.
Basically DPI does NOT apply to video screens; video systems know no concept of dpi at all or any concept of inches either. You should notice that the terms "dpi" or "ppi" simply do not appear in any user manual for any monitor or for any video board.
It's horses for courses, and PNG-24 would have a hell of a time trying to compete with JPG on a complex colour photo for file-size even if the JPG was saved at 100% quality.
Although without a doubt PNG-24/32 is the superior format for quality when using RGB images. Though for online photographs - as part of a web page design - you'd nearly always choose JPG mainly due to a small filesize; "lossy compression" trade-off with perceived quality when viewed as part of a webpage.