im interested in the smallest one in the pic, would i have any problems if i use that one even though it has a broken tooth?

Besides the missing tooth, the cog looks cracked at the base of the next tooth. The smallest cog has had it, and will self distruct if used very much longer.

You might be able to get away with it. It's in a worse location on the smallest cog, compared to if it were on the largest cog, because the small cog doesn't have many teeth to begin with.

As rjacob pointed out, it looks as if the cog is cracked, perhaps from impact. I don't think that it will self-destruct, but it probably won't shift well, the chain may hop, and it has the possibility of hurting your chain.

I'd try it - if it works well enough, ride it but watch it to see if the crack worsens.

On the other hand, you can get a new 6-speed freewheel for $20.

IF you insist on riding with THAT FW, set the high limit screw so you can't use that cog.
You're going to "gobble up" you chain which will then cause undue wear on the chain rings.

Usually Falcons eat up bearings before something else happens to them. They tend not to be long lived.
I'm not sure if a Shimano or SRAM replacement fits. Falcon MIGHT be a different "animal", with different threading???
Maybe someone knows for sure.

SRAM and Shimano cogs may thread onto the freewheel differently than Falcon, but for replacing the entire freewheel, nearly everything threads the same these days.

" nearly everything threads the same these days."
It's the "nearly" part I'm not sure of. I've never seen Falcon FW's for sale other than when installed on a Target or Xmart type bike. Since they are from China, I wouldn't be surprised if they had a proprietory threading or whatever to be used with a "Falcon hub". I know Park sells a removal tool for them, which probably indicates they are worth removing I wouldn't think anybody would remove one, if no "available" FW would fit.

You can try but it will likely skip a lot whenever you use that cog. I once broke a tooth on a commute to work and the chain would skip on that cog whenever I attempted to accelerate. Since it was a heavily used cog in the middle of the cassette, I replaced it.

Since the damage is on the smallest cog with the fewest teeth, and the metal is actually cracked (or torn?) on the tooth next to the missing one, I am thinking that crack (or tear) will only get stressed and larger.

I had an XMart bike with a Falcon freewheel. I replaced it with a new wheel with a Shimano freehub from the LBS for under $65.00. Got an 11 tooth cog instead of the 14, and got rid of the wobble the Falcon had.

That sounds like a reasonable solution.

One could also attempt to ride with the damaged high-gear cog by using it only at high speeds, under very light torque, remembering to downshift either to accelerate or as soon as one's pedal RPMs began to dip.