If I were you, ...

Camera: since u mention continous light, i think any decent cheap mirrorless will do. maybe a 2nd hand NEX? and since you are starting up business its better to start budget. today camera can do very very well at low light compare to camera 5 years ago.
Lenses: kit lens will do, i dont think u need a prime, because, many online shop just use handphone can snap good product pic too... dont over do it.
Lightning: yes, grab LED, and u dont necessary have to point it on your subject getting harsh shadow, u can point it up to the ceiling and let it bounce down giving u a very nice soft (multi directional) light. (applied same for video too)
Mic/voice recorder: not sure how u wan to video it, with yourself inside or you in the background? go youtube and check out "product review" video, from home review to professional review.
advise/tips: go check keyword "working from home, setup home for vlog" there are many youtuber actually earn money from youtube... by Vlog, especially gamers (seriously!) check out how they setup the room, to prevent ECHO, hiss and chips, your biggest concern in any indoor recording.

extra, depending how large is your product, consider a product softbox (white box to put product inside to shoot. it can help cutdown unwanted reflection from shinny product.