You can use your own products as long as they are professional-only or professional-standard. Scrubs, masks, lotions and traditional polishes don't have to be professional-only, just professional-standard as there are no professional-only brands for these. Everything else must be professional-only and not openly sold to the retail market.
Imagine it as if a hairdressing student did her entire course using boxes of hair dye from the supermarket - there is a market for retail products in the industry, but the quality and performance of those products is VASTLY different. For example, the monomer liquid we use ourselves is $220/Litre - but it performs accordingly. The retail brands can't sell something like that to the general population so they create cheaper, lower quality versions. Another example would be if you entered a car race in your family car instead of in a race car - there's a market for family cars and there's a market for race cars but their prices and performance vary greatly.
With your lamp, you must use the lamp that matches your brand of gel polish. Gel feels hard at 55% cured but isn't safe until it's 95% cured. Professional-only brands have labs that test the cure using their lamps and develop lamps that achieve 95% (or develop gels that cure to 95% in their existing lamp). Pigment and other formulaic changes all affect the cure as well, so every single individual colour cures differently and every colour has to be formulated to cure in that brand's lamp. Retail brands don't test as rigorously because they aren't being judged on longevity, they just need the general public to feel like they're doing what a Nail tech can do.
Once you decide on a gel brand, you'll need to get the lamp that matches it so that you can perform services safely.