I'm not sure what the law is about drop ceilings, but I think you must use Romex. I would use an inlet/outlet setup to follow code. A standard power outlet for about $1.00. Then the inlet like I linked to. This should meet National Electric Code and be fine for a permanent in-ceiling installation.
Well, I have long diatribes on ceiling mounts. I swear, swear, swear by the Chief RPMAU mount, or one of it's similar Elite mounts. The Elite mount allows for micro-fine tuned adjustments after it is fully installed. It does not come with a projector mounting plate or the ceiling connection. The projector mounting plate is purchased separately, or comes in as part of the kit. So, the RPM is a 'Elite' mount. The RPMAU is the Elite mount that comes with a locking mechanism (type-A) and a universal mounting plate (type-U). So, RPM A U or RPMAU.
You still then get a standard 1.5" flange for the ceiling, and a 1.5" steel pipe of whatever length you want. You can get 1.5" steel pipe from Home Depot as well as a 1.5" steel flange. Both are in the plumbing section.
The mount itself, I get on eBay. I search high and low for a good deal, and it can be had for under $100. The reason to get the universal plate with it is that it will work with any projector you may get in the future.
So, while something like this is ideal, and a fair price...
https://www.ebay.com/itm/Chief-RPMAU-Elite-Universal-Projector-Mount-with-Keyed-Locking/143080078800?hash=item21503cbdd0:g:7YAAAOSwUWtcL~9a
It's still pricey and there are better deals out there. If you can accept a color other than black, you can also get even better deals. Just a bit of looking. The RPM mount itself works with a ton of different plates, so sometimes it makes sense to buy the mount and the plate separate from each other, sometimes as part of completely different purchases/products. It can be weird.
ie: For $28 you can get the RPM Elite mount with a pole (too long) and a ceiling plate shipped to your door.
https://www.ebay.com/itm/Chief-RPM-Series-Projector-Ceiling-Mount/123485713744?epid=19015178216&hash=item1cc0527550:g:2jUAAOSwEKpb5dIr
You will still need the plate that attaches to your projector. That can be a problem if you want the really good SLMU universal bracket...
https://www.ebay.com/itm/Chief-SLMU-Universal-RPA-Interface-Bracket-Black/192669103686?hash=item2cdbf93646:g:CcwAAOSwWhVbqTJn
So, not really cheap, but about $120 or so for everything but the pole you need.
But, you can find packages with the plate for a good price...
https://www.ebay.com/itm/CHIEF-RPA000-4GH8-A-V-Ceiling-Universal-Projector-Mount-Assembly/143079856849?hash=item2150395ad1:g:~T8AAOSwjMtb8GN-
That only has 3 legs, but you can buy a fourth leg, if you want/need it for about 20 bucks or so.
Better deals?
White, only 3 legs...
https://www.ebay.com/itm/Chief-RPMA000W-RPA-Elite-50lb-Capacity-Projector-Ceiling-Mount-Bracket/192457127322?epid=2253277442&hash=item2ccf56b59a:g:jIgAAOSwEoJahcjr
This one is the smaller bracket (up to 25 pound projector weight) with the universal bracket in silver...
https://www.ebay.com/itm/Infocus-CHIEF-MSP-IFRSMAUS-52524/331987559366?hash=item4d4c0023c6:g:wiUAAOSwOyJX7-km
Wrong mount, but good bracket, reasonable money. Pairs well with the $28 mount I linked above...
https://www.ebay.com/itm/Chief-Projector-Ceiling-Mount-Universal-Epson/173614620118?hash=item286c3cddd6:g:mk4AAOSwTxpb17bF:sc:USPSPriority!20152!US!-1
Of all of these, if you need/want black, then the first link I gave for the $30 mount and then two links later for the $30 plate are the best deal I've seen on eBay right now. I have found other good deals in the past, and they come up from time to time, but they often go quick and when others ask and I see a good deal, I recommend it. So, that's what we have on eBay right now that I can find.
If you are handy, you can modify a 'custom' mount to fit your projector. Just a few bolts and a drill and perhaps a hacksaw. No modifications to your projector. Not sure how handy you are.