Last I checked (which admittedly was a couple of years ago), all-in-one printers had far inferior print quality for photos than a dedicated photo printer. Using a high quality photo printer for day-to-day text printing is nonoptimal, too, as it drinks up much more expensive ink than a laserjet type black & white printer, and you'll be constantly switching between high-quality paper for photos and low-quality paper for cheap everyday printing. The standard approach is to get a high-end photo-quality color printer (Epson, HP, or Canon) for photos and a cheap laserjet for text printing. If you want really good print quality, you won't get it with an all-in-one (unless they're made amazing improvements recently, which is always possible...).
Regarding inks, if you don't have a substantial amount of experience with color management issues and futzing with printers, it's safest to use the OEM ink the printer is made to work with rather than trying third-party inks (which do exist, but are more difficult to deal with if you don't have much experience). If you decide you need archival inks, then get a printer that comes with them and use the ones it's made for; if you prefer dye inks, ditto. Things will just work much easier that way.
Archival inks with archival paper will last for many years. Dye inks often (not always, but most of the time - depends on exactly which inks you're talking about) have a bit more vibrant color and a bit blacker blacks, but can fade or color-shift quickly. If you want something to look at for a couple of weeks, dye inks are fine; but if you want your prints to last much longer, you should go with archival inks.
If you're relatively inexperienced at making high-quality prints, it's safest to stick with papers made by the printer's manufacturer, or with papers recommended by people who have used them with your printer. In any event, you'll want to make sure you have a printer-specific/ink-specific/paper-specific color profile available for whatever combination you're using. If you're a beginner, I'd recommend choosing a printer and then looking at samples of the high-quality papers made by that printer's manufacturer (for example, Epson encloses print samples of a number of their papers with some of their printers) and choose the one(s) you like the look of. If you use OEM papers, profiles for those papers will be included in the printer software (at least for Epson - I would be very surprised if the other major manufacturers would be any different, since they want to sell you their papers).
I wouldn't worry too much about specific DPI or resolution numbers. There's far more to excellent image quality than that. Read some printer reviews (from serious photography web sites or other media, not general consumer media) to choose one; you can't just go by a couple of simple numbers.
Of course, I'm assuming you want something you can be proud to hang on your wall (the prints, not the printer ) . If you just want something to print snapshots to show your mother, this isn't the right place to ask, and ignore this post; check Consumer Reports instead.
Does this answer some of your questions?
Lisa
