There are many ‘barns’ devoted chiefly to jump racing, but 99.999% of those also do foxhunters (directly linked to steeplechasing - look up 1752 St. Albans/St. Buttevant for ‘the story’) and/or flat racing.
Most stables like this ^^ aren’t working it as ‘a business’ per se, and losing money is part of the game. There are the few like Rowdy Irishman that keep ‘the dream’ alive of making it big with a fairly random horse, but most are owned-operated by rich patrons (for lack of a better term - that polo word works nicely) who employ talented, skilled and highly experienced horsemen to run the operations
You don’t look for a horse to run over timber - they just appear. As I said, sincerely, most jumpers have run on the flat, and when they’re out of conditions, they (sometimes) turn to jumping. You’d of course want a nice shoulder, good bone, short cannons for soundness (jump horses can and do run well into their teens, with some Hunt Cup winners at age 15), relatively low set hocks, etc. but that’s all part and parcel for any talented racehorse.
Many jump horses (you’ve GOT to stop thinking of it as ‘timber horses’ - that’s a thing, but it’s not how one views steeplechasing, at all) were not very good on the flat, but the good ones were good. You used to be able to win a race with a fleet foxhunter (see St. Albans/1752) but those days are long gone. You better have some pedigree and speed to win on today’s US circuit.
There really isn’t a ‘type’ you can identify for hurdles/timber/flat, b/c good conformation=good conformation, though if you must, you’d think of a timber horse as a bit bigger (not always) and a bit older (not always), maybe a bit more quarters but, sincerely, I defy you to look at John’s Call (G1 sw on the flat) vs. Saluter (timber champion) vs. Skip Away (hurdle champion) and identify any particular conformation quirk that made them tops of their venue.
Training schedule is much like the flat track - coming from ‘time off’, couple weeks jogging, couple weeks shorter cantering, several weeks 2 miles galloping a good lick - around 400-450 mpm. After that add in 1/2 mile 2 minute lick work (2 minute mile - approx 800 mpm), day off, then gallop another week, add another work, a tad longer, etc. with 4-5 works leading up to a race.
Many trainers use training flat races at the jump meets (in the US) or bumpers (UK) as ‘works’ which pretty much count as 2 home-works, and keep the miles off the horse’s legs while tightening them for their jump races.
Timber horses do NOT run over hurdles as ‘prep’ races when running over timber. A few of the cross-country racehorses run primarily over timber but add maybe a point to point hurdle start to remind the horse of the brushing-through jumping style for their varied cross country racing.
99.99999% of jump horses train ‘on the farm’, so for variation, add in a hack day or a hunting day or a roadwork day or a flatwork day (in the arena, just like a sport horse), and an occasional walking day if needed due to weather or whatever.
A typical ‘day’ would be perhaps morning turnout after breakfast while stalls are done, morning gallop or work or exercise or whatever, bath/hotwalk if needed (though several trainers I worked for simply ‘walked the horse home’ from the on-farm training track (a good half mile), hosed them off (cold water all seasons) and chucked them back out in the pasture to finish cooling down.) (And before you gasp, two of them were Eclipse winners … 
Horses will often go back out in the paddock or pasture the balance of the day; some come in at night, others don’t. My biggest influence trainers never kept horses inside; I pattern my own (miniscule and occasional) jump training after that - and I’ve trained 4 Va, circuit overall champions, so I guess it works pretty well … )
As to a ‘rider’, they get on horses to gallop in the mornings, school horses over fences (young horses get more work over jumps, naturally; veterans often just get 1-2 tuneups before they run first time that season; zero after they’re running that year.) Most riders do babies in the afternoons or something else in the horse business - I suppose some of the top jockeys only get on horses in the morning and don’t have to do young horses or exercise hunters in the afternoons but I’d say a majority percentage of ‘riders’ do something else at least part of the rest of the day.
Off season, they’re 99.9999999 turned out, as in, muddy and fuzzy and messy and outside 24-7 almost always with buddies,
