If the height between the signal trace and the ground plane is h, a fair rule of thumb is to keep all potential perturbing features at least 3h away from your traces. If you can manage more separation, that's even better.

Also, if the trace length is less than 1/10 wavelength at your frequencies of interest, determined by the rise and fall times of your digital signals, remember that it probably doesn't matter much what you do. That's a trace length of 1.4 meters at 10 MHz or 14 cm at 100 MHz. If your sketch is showing through holes spaced at 0.1 inches, it looks like your board is less than 1 inch square and you could get away with well over 100 MHz signals without worrying excessively about controlled impedance and careful terminations.

Edit

This is not to say you should totally ignore good design practice and get rid of your ground plane or run traces across slots in the ground plane, as indicated in comments below. Also, the distance values above (1.4 m and 14 cm) are corrected from my initial answer.

The 50Ω standard is basically just convention. There are various stories about how 50Ω came to be chosen. The article Anindo linked is good. There is also The History of 50 Ω or There’s Nothing Magic About 50 Ohms. But the long and short of it is that it is a compromise between low attenuation and power handling.

But it became the standard impedance when designing for transmission line applications way back when. When an IC datasheet says you need to design your PCB traces with a controlled impedance, then you're designing to compensate for transmission line effects. If the impedance of the trace is matched to the output impedance of the IC or source, you reduce the possibility of reflections which would lead to standing waves on the trace and cause all sorts of headaches. Since the designers of the IC are designing with transmission line effects in mind, and since 50Ω is commonly used by convention, the 50Ω standard proliferates.

But 50Ω is by no means special. From this paper on controlled solutions by Advanced Layout Solutions:

Within reason, the absolute impedance value chosen is not normally important, providing it is controlled along the entire length of the line. Other constraints in a design often dictate the impedance for us; it may be chosen based on a design specification (e.g. 65 Ohms for PCI) or chosen to reduce current (a high impedance). It will generally be between 45 and 80 Ohms due to typical material geometries, and if the signal changes layer then the trace geometry should be adjusted as necessary to maintain a consistent Zo.