Yes, you should go 1.5-2.5" up in seat size from western to English. I ride a 16.5-17" western and take a 18.5-19" English saddle, depending on the seat shape/type.
English saddles aren't usable when they get really old and beat up. Often the repair costs exceed the value of the saddle. So, very old or beat up English saddles usually end up in the trash or in someone's living room as a decoration.
When looking for an English saddle, go with a brand name only. No-name English saddles are often very poor quality and can sore the horse. They also usually put the ride in a poor position, making riding uncomfortable.
Stirrups and leathers are sold separately in most cases because they are available in many stizes and "types". It's personal preferance as to what type you like. I prefer jointed irons because my knees are "iffy" and short English stirrups put extra pressure on them. The jointed irons help alleviate some of that pressure.
If you're just getting in to English or wanting to try it out, I'd suggest finding an All Purpose saddle of a good brand. Good AP saddles will be comfortable, put you in a decent position, and will be a bit more secure than a flat close contact/jumping saddle. I would look at brands like Stubben, Wintec, Thorowgood, Courbette, Crosby, Collegiate, Pastene (older Argentine brand), Avante, and any other saddle that is "Made in England" or other European country. Some Argentine saddles are good, but some are not. Borellis are okay, but only if they're in good condition.
If your horse takes semi-QH bars, then a medium tree will likely work. If your horse takes QH bars, then a medium-wide tree will likely work. If your horse takes Full QH bars, then you'll likely need a wide or x-wide tree. The changable gullet saddles work on most horses that aren't terribly flat backed or super wide barreled.
For stirrup irons and leathers, average adult women take 4 3/4" irons and 54" leathers. If you have large/wide feet, you may need a 5" iron. If you wear "short" or "petite" length pants, then I'd go with 48" leathers. If you have really long legs, then you may need 58" or 60" leathers, but only if you're very tall, 5'10" or taller or with an inseam that measure truly 36" or longer.
Leathers and irons should be good quality as well. Cheap leathers will stretch and rub the saddle, and cheap irons will rust. You do NOT want chrome plated irons. Look for stainless steel irons or aluminum. If you can't afford good leathers (usually $75+ for a pair), then I'd go with Wintec Webber or Wintec's synthetic leathers. Thorowgood also has some nice synthetic leathers.