The oil cooler is there to keep the engine bearings happy. This is something that becomes more important as the engine ages, not less important. So, make sure the oil is cooled. Do not bypass the cooler in the rad unless you are adding an equally sized or larger cooler external to the radiator.
New aluminum/plastic radiators can be had online for about $100. And after pricing them online, you might want to see if a local radiator shop will match price for you. I found one that would in my town.
Re the cooler lines: If the oil pressure relief valve is working as it should, the oil cooler line should not normally see more than about 60 psi. But, we often see higher pressures (80-100 psi) on cold startups until things warm up a bit. Speaking of which, engine oil can get pretty warm, so, if you add any hose, make sure you get hose that is rated for hot oil. Hot oil will make quick work of regular rubber hose. You should be able to use properly sized hose barbs and gear clamps to effect a temporary repair. But, you might want to look into the cost of replacing the oil cooler line at the dealer or with an aftermarket equivalent. You may be surprised at the cost. For example, a new set of transmission cooler hoses with the proper bends and fittings on the ends for my '95 cost about $55 at the dealer. Perhaps the oil cooler hoses are similarly priced.
Re the coolant in trans: Coolant in the trans fluid will destroy the transmission. The quicker you get it out of there, the better. Maybe you can save it. Coolant is heavier than oil and should drop to the low spot in the trans which hopefully, is the pan. In this case, you may want to consider draining the fluid when the trans is cold. Because I am not a fan of flushing, I would drop the pan and change the filter, reuse the steel/rubber gasket (if it is still there), then do it again in a few hundred more miles. You want the fluid to be clear red, not milky pink, tan or orange that would indicate an emulsified mix of coolant and oil.