Stand-alone filters are basically marketed in two capabilities -- big stuff or small stuff. Big stuff is visible dust (specs you can see floating in the air when sunlight hits it). Those filters are aimed at people who want to reduce how often they need to clean what collects on surfaces. Small stuff (many allergens) is mostly too small to be very visible when it's airborne. Filters aimed at allergy sufferers are designed to remove the small stuff.
BTW, dust mite allergy is mainly to their waste, which is a lot smaller than the mites. Your idea of reducing their food source with an air filter won't really do that. Their food source is your discarded skin cells that fall on the floor, furniture, and other surfaces. The skin cells don't stay airborne, and if they did, that would keep them out of the reach of the mites, anyway.
The mite poop is actually sitting safely out of the way on surfaces until you disturb it by kicking up floor dust, sitting on upholstered furniture, or dusting or vacuuming hard surfaces. You can make a huge difference by regularly vacuuming the floor and seating surfaces with a HEPA vacuum (a one-time cost and some are not all that expensive), and using an electrostatic microfiber cleaning tool (like Swiffer) to dust hard surfaces. That will cut way down on what gets airborne in the first place.
The stand-alone units are generally sized to clean the air in a single room. They don't move enough air, or move it in a useful pattern, to circulate the air of multiple rooms connected through doorways. A huge open floor plan that is a much bigger space than the unit is designed for will also limit its usefulness.
The general principle for allergy relief is to use the filter where you spend you time; the bedroom at night is usually the first priority. If you spend much of the day in a particular room, move the filter there during the day. If you're between a lot of rooms during the day, just pick one and let it work on the air in general.
The filters designed for allergy suffers are mostly one of four designs.
The simplest is something you could actually make yourself. It's basically a fan in a box that holds a high-end, "allergy-rated" pleated furnace filter. The fan circulates the room air through the filter. When the filter gets dirty, you replace it. These have a cheap initial cost but replacement of the filters every few months can get expensive. They won't be as effective as a HEPA filter, but are likely to be a big improvement over nothing. They will capture particles of all sizes.
The HEPA filters have a washable prefilter to catch the big stuff and keep it from clogging the HEPA media. Some of the HEPA filters have media that can be cleaned a few times. These units are very effective, but not cheap to buy and you will need to replace the HEPA media every so often, which can be expensive.
Tabletop electronic air cleaners use metal plates charged with a high voltage to attract and hold particles as a fan pulls air past them. They're good but not quite as effective as a HEPA filter for the smallest particles. There's nothing to replace unless they have a prefilter to catch big stuff. Those are typically washable, and inexpensive to eventually replace.
The metal plates do require regular cleaning. Big particles being captured will produce a little zapping sound (probably the reason for the prefilter). They can produce tiny amounts of ozone, but generally not enough to be noticeable.
Certain ionic air cleaners clean room air without leaving molecular-sized crud electrostatically bonded to all nearby surfaces. These are similar in principle to electronic air cleaners and move the air either with a fan or the wind created by generating ions that are attracted to metal plates. Sharper Image is a big seller of this type. They are absurdly over-priced.
They move much less air than the other types of cleaners, moving it relatively slowly through the collection system. They are very effective for the smallest particles (odor removal), but they don't do much with the bulk of the dust volume, which is made up of bigger stuff. Don't waste your money.