What are the effects of the different design parameters when building a linear motor?
Good question.
You earn a PhD by the time you get a half good answer :-).

More detail about what you are wanting to achieve wouldhelp muchly. Stroke, force, speed, ... . Appliaction?

Too much to cover at present.  This is just a quick canter and their should be others more expert here.   As well as "learned papers" you can get a good idea by using eg Google image search using eg linear motor as search term, looking for manufacturers "strutting their stuff" re new models, advantages, capabilities etc and seeing what they say. 

But:

Air gap is crucial.
Magnetic field varies with inverse cube of distance at a distance (really) due to interaction of two poles working in inverse square. (FWIW few people realise that this is why what happens happens). Magnet shape and thickness and proximity to other magnets affect the field. At distances up to 5 or 10 magnet thicknesses if you want to see what the field is at a point you are best to use one of the many free and far from free simulation programs. 

For practical purposes for DIY linear motors alternators etc you can use the rule of thumb that the magnet is about as effective as it's going to be out to about half its thickness. eg if you use a really top grade "rare earth" magnet (details later maybe) you can get around 1T+ at half it's thickness. So for "ironless" coils if you are using 10mm thick magnets then the majority of your coil wants top be no more than 5mm away from the pole faces so if you have a 1mm airgap then you can have 4mm thick coils.
Which is why people use laminations and why ironless construction has very flat coils. Once you add laminations in the windings you "extend the field" at the cost of more mass, eddy current losses, magnetic saturation, saliency (cogging) unless you are very good at mechanical design, and other secondary effects. 

About all the rest is standard theory at a first approximation. You can find papers on motor design, ampturns, conductor resistance (which affects heating and current you can provide to get the amp-turns that do the work. If you use laminations you are into the wonderful world of saturation, BH curves, losses, fringing effects, ... . But it's all available in the standard motor texts. 

As a simple simplistic starting position, to get maximum motor force you are trying to use the strongest magnets you cn get, smallest airgaps, flattest coils, and maximise amp turns. Amp turns are affected by the copper (usually) you can fit in the available space and the current you can apply and the heating that results.