I’ve been reading and following these comments now and I’m both impressed and horrified at the same time.
As a mechanical engineer, product developer and developing air purifiers specifically for 5,5 years in combination of filter expert till just 6 months ago. I now work in another industry. You can ask me anything and I’ll do my best and be as unbiassed to brands as possible if you want me to.

There are a few things you need to know for your own safety if you do DIY and want to reduce costs:
Fan and filter are the heart of the unit. This is where you put the money even if you want to do a temporary solution. The housing can be cheap, but please be smart and never leave a DIY unit alone since electricity is involved, they are not safe. Once you’ve bought all this equipment, you’ll soon realize you have put out a cost of a reasonably good air purifier.

-	You need to have a good amount of space between the filter and the fan to cut pressure drop – otherwise you will overload the fan, you get a high electrical bill and it becomes a fire hazard or it will break down = your money is not well spent. This is why you see some units on the market looking hollow – the space is there for a reason.
-	Fans. Look for a strong motor that can handle air resistance. Regular fans are not made for resistance and will break, stop (programmed) or even start a fire. Also look at the airflow rate and the safety regulations of it. The higher airflow rates the better. When it comes to fans, you get what you pay for.
-	If you use a weak fan and a thick filter you will only clean the air closest to the fan, not the actual room air = money not well spent unless you use the purifier right in front of your mouth/nose.
-	HEPA filters are actually not that great... There are newer filter types on the market that are less dense but still not as effective. HEPA can be too thick/dense for the fan to be able to pull air through, or the air doesn’t have high flow enough to clean more than just the air next to the purifier. You need to have circulation. Especially fans that are made to only move air without resistance, the motors are not adapted for this. If you have a strong fan you can use HEPA (Merv) since it can power through. Also, if the filters are too thick, air leakage will occur, motor will sound horrible, and the air will pass on the side instead of through the filter = noises, whistle sounds and less clean air. Yes, duct tape is good here if you build it yourself.
-	Put the filters in front of the fan (before fan) to spare the life of the fan – the dirt will not accumulate and destroy the fan over time. To get (collect) dirt in the filters is the goal. Some fans can handle dirt passing through and then you can have the filter after. However, if you just buy a regular air fan in a regular store, I don’t suggest having the filter after the fan (air flow direction).
-	If the filter gets super dirty quickly, you can vacuum or gently whip it outdoors to recover some lifetime. However, after some time (roughly 1 year in normal use) it will be clogged no matter what you do. There are no filters or technologies better than particle and carbon filters available today. The 6-months to a year lifespan is actually accurate in normal conditions. I’ve been doing these tests in China, India and Europe and US several times in different conditions (both lab standard and normal homes) and with different units/tech/brands etc and can confirm particle filters + ionization is the best tech available and the filters last 4 months-a year depending on local conditions. The filters are usually functional after a year as well, but with very low efficiency.
-	Pre-filters are great since they take on large particles like pollen or ashes before it hits the main filter. Just make sure it’s not too dense.
-	Fully made air purifiers are tested by 3d party institutions unbiased to payments or commercial, certified and heavy controlled before released on the market. You can check consumer reports, AHAM and California Air resource board (ARB) or Chinese GB-standard websites for different type of tests and results if you want to know which ones are really, really good. Some complain that they are in “lab environments” but it’s still really good products that does the job even in real life conditions.
-	If the unit you want is not in their lists of tested products, it’s a warning sign that the product is not good, or might not even qualify as an air purifier… It’s a surprising large number of big brands that make terrible products but are good in marketing or simply rely on their names.
-	Fully made air purifiers have fans developed specifically to match the filters resistance and won’t break down after a few months or start a fire if put under heavy resistance (clogged filters). The filters last longer since its optimized for air treatment in home environments. Usually other technologies are a part as well such as ionization or UV or similar. That’s why they are more expensive. They are super important in a development of air purifies. Some of them also have IP44 certified fans meaning they can work during humid conditions without breaking or causing short-circuit/burn outs or worse.
-	These DIY things work, but are usually not cheaper in the long run and are not as effective as the real purifiers. If you suffer from fires, smog, allergies etc I highly suggest getting a real purifier incl filtration+ionization since those things are very harmful, especially for children. If you just like to temporary fix something these DIY will work if you do it right.

I hope I could answer some of your questions.