Pat's right on the money. I'll add:
When checking the ring gaps, if they are too large it generally won't cause a problem unless it's WAY too big. Rule of thumb is .004 for each inch of bore diameter, i.e. 3" bore = .012 ring gap. On that same bore, if the ring gap is, say, .020 it might not be perfect but it's not going to hurt anything on a non-race motor. Now if it's .060, I'd be concerned. That would tell me it's very possible the bore is too big (improper machining) and I'd look REAL carefully at the piston skirt to bore clearance.
That's a critical clearance, by the way. Too tight = seizure, particularly in air cooled engines, too loose = at least piston slap when the engine is cold and in extreme cases breaking the skirt of the piston! Check it by either measuring the bore diameter and the piston diameter, subtract piston diameter from bore diameter and there's your measurement, or use a selection of feeler gauges as a 'go/no go' type of thing. For instance, if the skirt clearance is spec'ed at .008, then you should be able to insert the piston in the bore with a .008 feeler gauge between the piston and bore but if you try it with a .010 you should not be able to fit the piston into the cylinder. If the .010 fits it's time to go talk to the machine shop.
If the ring gap is too tight, yes that is something that needs to be corrected. Going by our same example above, if the spec is .012 and the real gap is .006, as the rings heat up and expand the ends will butt together. As the ring continues to try to expand, it forces them way too tight against the cylinder wall. Result: seizure. If the gap is too tight, use a small file or a knife sharpening stone to trim the end of the ring.
If the oil control ring is the three piece design with an expander (very likely: one piece oil rings are nearly extinct), the rails (the thin ones) usually have a pretty large gap compared to the compression rings (the thick ones). When installing the expander (the wiggly spring lookin' ring), be VERY sure the ends of the expander butt up to each other and do not over lap. If they overlap, the engine will burn oil like nobody's business and can even seize since the rails don't fit properly in the ring groove.
One other thought about that thick base gasket: how much thicker is it? Reason I ask, if the head was resurfaced, say, .010 that's going to raise the compression. If the thick gasket is .010 thicker than stock it puts the compression back pretty close to stock. I'd also look into some Loctite red gasket sealer, I believe it's their no. 518. That stuff works wonders on thin paper gaskets and adds no thickness.