jimmuller 

Without a close examination one can't be certain what you mean by this, but some freewheel or cassette teeth are slanted or shorter by design. It helps them pick up the chain better in a shift. Which is to say they ain't broke, they're s'posed to be that way. You'd have to hit them with a hammer to bend them sideways so a funky chain surely wouldn't hurt them except make them wear into odd shapes. Other than the surface rust on those cogs I don't see anything wrong with them. They look to have no wear at all.

Perhaps modern chainrings have an occasional shaped tooth also for the same reason, but my experience doesn't extend much into modern chainrings so I'm only guessing.

I'm with that wandering architect and the guy who doesn't glide. Lube the hubs and bottom bracket, get a good chain, good tires, and alloy wheels, then ride it. Skip the wheels for now if your budget is tight. A bad chain will eventually wear out the chainring and cog teeth (but that takes a lot of miles). Plus it has lots of bearing surfaces which create friction if they aren't clean and free. And a sloppy chain shifts poorly. Once you've ridden enough to judge the rest of the bike you can decide whether you like it well enough to take it further.

One more thing. You asked about tire size. People argue about that all day. I have 25mm (1") Pasela (TourGuard version for flat protection, more expensive) tires for two of my road bikes, but I've got 28mm (1 1/8") on a commuter road bike, and 25mm Gatorskins on another road bike, and much more expensive 23mm Vittoria Open Corsa tires on my best bike. Can I tell the difference between them? Oh yes, especially the Vittorias, but some of that is probably the bike differences too. If your roads are smooth, go narrower, if they're rough go wider.