I never had much sediment in the bottles of sparkling wine I've made, just a light dusting on the bottom. I don't use fining agents, I just let the wine clear, rack to the bottling bucket, prime & bottle. I just pour carefully & leave the last little bit in the bottle, just like I do with home brewed beer or cider.
As to bottling, always use the proper bottles. Use champagne bottles, they're designed to withstand the pressure, regular wine bottles are not. I've always used the plastic stoppers & wire cages because they're simple & easy.
I just gently tap the plastic stoppers in with a rubber mallet, put a wire cage over it & use the little twisting tool to twist the wire. It might take you a couple of times to get a feel for how tight to twist, but it really is simple & easy. The twisting tool is essential & will save your fingers from all the twisting.
You can get everything you need from virtually any homebrew supply shop, even order from an online store & have it all delivered to your door. IMHO, bottling with the plastic stoppers & wire cages is easier & quicker than using a wing capper & crown caps.
Regards, GF.