I'm sure she's strong, but it's hard to push down with more force than you weigh, unless you really pull on the handlebars. My point is that it's unlikely that she's putting down that much more force than some of the larger men.
Any of the major manufacturers (Trek, Specialized, Giant, etc) have quality bikes, see your local bike shop (LBS). Expect to pay around $1k for a new hardtail, $2.5k for full-suspension.
For good quality component you want Shimano Deore or higher (SLX, XT, XTR). Once you get above Deore, reliability and quality don't improve much, you get more features and lighter weight. Deore and SLX are a good sweet spot for Shimano components.
Some of it is likely a combination of technique and poor or bad components. You want to ease off the pedals while shifting, and soft-pedal until the shift completes. Better components make the bike shift faster, so you can get back to pedaling faster. Shifts should be a quick click on both up and down shifts so it takes less than a pedal revolution.
Anticipating your shifts is very helpful, a strong kick or two will give you enough momentum to soft pedal while the shift completes. Shifting as your cadence drops, before it bottoms out is critical.