Greetings and Salutations. I remember this question now. I didn't bother responding figuring the more Rifts experienced players would handle it. Anyways, there are a few things. As stated above, Area Effect abilities can work, but I'll focus on the abilities itself. I'll address them one at a time.
Art of Vanishing or Sun Shih K'an Chien Chih (Rifts Japan, page 195): First, the character needs a really good environment to pull this off. If it's really dark and lots of stuff to hide behind, sure, it's going to be tough. But in good light and no place for him to really move, even the level 6 character may only have a 50% chance of pulling this off. If he's lucky, good for him. If not, then he just left himself open for attack (trying counts as an action). Now, this ability will let the character hide and a chance to prowl up and attack from behind, but that also requires prowl checks and the character to leave himself open. Putting his back against a wall or moving into a corner will make it really hard for the ninja to sneak up.
ARt of Mystic Invisibility or Chi Zoshiki (Rifts Japan, page 196): First off, there is a saving throw. It's rather high though (19 or higher, and as best as I can see no bonuses). That's only a 10% chance of saving, but there's still a chance and this can help. Second, while invisible the ninja cannot attack or perform skills. This means the character must turn visible to attack (as you know). As a result, an alert character should be able to defend. Granted, most characters don't have eyes behind their heads so a back attack will still work well. Of course, having a friend watch your back (standing back to back) makes sneaking up behind them really, really, really, really hard. You can't attack what you don't have access to. And, just to clarify, the ninja turns visible, then attacks. It's not attack, hit, and then turn visible. A lone individual is still likely in serious trouble, but this is one of the reasons why team work should not be overlooked. Additionally, having just two people around forces the ninja to spend four times as much energy. Get 9 there and, in order to fool them all, the ninja is burning P.P.E. fast. He may (if really, really good) be able to get off a few shots and then go back invisible, but he'll be out soon so he better think really hard about what his plan will be. And, truth be told, if he comes up with a really good plan, I'd likely let him get away with it (good plans should be rewarded). Also keep in mind, this ability can only be used once per melee round. That means if they see pass their save or he attacks too early (or even too late, because cost is per melee round and they get a saving throw per melee round) that he won't be able to use the ability again until the next melee. The character must spend at least 1 minute to just get off 4 attacks this way, and spends 4 P.P.E. if one on one, 16 P.P.E. if 2-8, and 48 P.P.E. if 9 or more. During this minute the victims can call in reinforcements, make plans and actions such as standing back to back, getting against a wall, or other action to help negate remaining a victim.
Dealing with these two powers together: Hmm ... well, truth be told, I don't see them both being used together really being that hard. After all, there isn't much overlap. For the most part, it's one or the other (but it is a good back-up if one fails). Mystic Invisibility can be done standing in plain sight, so you don't have to hide for it to work, so Vanishing doesn't play much part here. And if you've successfully Vanished you don't much need the Invisibility (unless you go out in plain sight again in which case Vanishing is no longer relevant). While these moves may be killer in a one on one battle you get two or three NPC together and suddenly these moves are an aid, but not the win or loss of the battle on their own. My best recommendation, and I can't stress this in enough games, is to use teamwork. Though as said above, using Area Effect tactics can work as well. Hopefully some of this helped. If not, then my apologies for wasting your time. I'm not an expert on Rifts in general, so if I've missed some basic problem why this is more deadly than I've given it credit, please let me know and I'll try to see if I can address the problem. Please have a nice day. Farewell and safe journeys.
P.S. To Lost One's question, probably not. It's a mystic ability (To befog men's minds), so if you fail you pretty much failed to see the person, end of story.