... can I use the old lenses with this new model?

Yes. The 7DMkII is a "crop" body, so it can use both EF and EF-S lenses. It's only if you move to a full-frame body like the 5D/6D bodies that you can no longer use EF-S lenses.

And if so, how much quality would I be losing by doing so?

Since both of those are considered to be entry-level kit lenses, you're unlikely to be losing much quality, because there's not a whole lot of quality to lose. ;) However, given the processor and sensor improvements between the 300D and the 7DMkII, you may actually see far more flaws. The 6MP to 20MP resolution increase, in particular, may point up a few more lens shortcomings.

In addition both of those lenses as entry-level lenses have been replaced in the intervening decade. The current 18-55 kit lens is probably the fifth or sixth iteration of that lens, and now sports both IS (stabilization) and STM (a step focus motor). And the 75-300 III has been replaced by three lenses: the EF-S 55-250 IS (cheap entry-level), the 70-300 IS USM (mid-grade; improved optics and updated IS over the 75-300 III IS USM), and the 70-300L IS USM (pro-grade; improved optics over the 70-300).

And to top it all, neither of your lenses is particularly well suited for sports shooting, which is not an entry-level subject. Sports tends to require autofocus speed. The 7DMkII can certainly deliver on the body side of the equation for that, but you will most typically want a ring-type USM lens for that on the lens side (with video, STM can be more useful; while a little slower, focusing will be smoother). And if the sports are indoors or at night, you'll probably want a lens with as wide a max. aperture as possible. And good telephoto zooms that do that on the Canon side of the fence tend to be white and cost four figures. Sports and wildlife are two areas where the gear costs are not modest.

IS (stabilization) may or may not be key, depending on how long your lens is, what the lighting conditions are for the specific sport, and whether you plan to shoot videos or stills. With video, IS can help smooth out panning. With stills, sports typically requires higher shutter speeds to "freeze" the action, so you may be up and over 1/focal_length (a rule of thumb shutter speed target to mitigate handheld camera shake blur; i.e., with a 300mm lens, you want to be faster than 1/300s) most of the time, anyway. If we're talking soccer, a 75-300 may be ok. But indoor hockey, swimming, or basketball, probably not.

It seems that the Φ67mm difference might be important also?

Well, it simply means you can't use the 58mm filters you already own without vignetting. You'll have to buy 67mm filters, but with a step-down ring, you could use 67mm filters on your other lenses without vignetting. Of course, if you end up getting a white L lens, chances are you'll need 77mm (or larger) filters for it.

If you want to decrease repeated filter purchases, purchase the largest size you need, and then the appropriate step-down rings. Unfortunately, the larger the filter size, the more expensive filers get, too.