*I'm not sure if I want to use "down" or "platz", any suggestions or comments on which word I should use?*
It really makes no difference, just pick one and use it consistently. You can also train formal commands with one word and more relaxed commands with other words if you want, as long as the criteria for each are different, and you are very consistent about reinforcing those separate criteria. Examples might be a formal recall to an automatic sit in front, (such as you'd use in competition), vs an informal command you'd use around the house, where you don't require any particular position or speed you just want your dog to come towards you, or an attention heel vs simply walking on a loose leash without pulling.
The only issue is talking to my boyfriend about how I want Varick to be trained. I use the word "yes" as a replacement for the clicker. This lets him know that what he did was right, so he'll most likely do it again (positive reinforcement). And I don't have the toy in sight while I'm training. I also don't want him to only lay down from the sit position, I want him to lay down from stand.
My boyfriend doesn't use my "cue" word (yes), so I need to tell him that this is the word he always needs to use. If he uses something else, wouldn't that be confusing for Varick? He always has the toy right in front of his face (so Varick is looking at the toy, not in his eyes).
Actually, "yes!" is a marker, your cue
would be the word you use (or the hand signal) for the behavior.
I do use both a verbal marker and the clicker and that doesn't seem to confuse my dogs, but I would discourage using more than one marker word, as it could definitely be confusing. As David said, it's fine to lure for awhile, but I like to lure only as long as absolutely necessary, and then use an empty hand for the lure motion and treat from the other hand (if you're training with food rewards), or get the toy out of sight if you're using play as the reward. Later, I fade the lure motion to something more subtle and that becomes my hand signal for that behavior, but I want my dogs to respond to my hand movement, not the treat or the toy in my hand.
With "down", it looks like this: I start by luring with a treat, taking my hand all the way to the floor, marking and giving the treat to the dog as it drops into a down. A few repetitions of that, and then I try the same motion with an empty hand, but still marking and rewarding. Gradually, I fade that big movement of bending over and putting my hand to the floor to a single finger pointing down, while I remain standing.
It can be difficult if your boyfriend doesn't want to respect your wishes with regards to how Varick is trained, but hopefully you can explain to him how important it is to use the same marker word, and the same criteria for each behavior that you are.