It's Jordan who said they got their souls from the soul pool, and that unlike the souls used for Nyms (that can later be reborn as humans), those that got a life as Shadowspawn are so tainted they can only be reborn as Shadowspawn (which unless Moridin one days manages to win for the DO and bring the end of Creation means Aginor holds the palm for committing the largest atrocity of all, condemning billions of souls to lives as Shadowspawn and eventually not to be reborn ever again when the Shadowspawn become extinct. As we don't know if new souls are ever "born" or if the number of souls is finite, it could also mean what Aginor has done with the Trollocs could have meant in the long run the extinction of all life but Shadowspawn...)

My guess where the OP or TP came into play for Constructs is that it's used to splice and reorganize their genes and bind their DNA. It may not be possible to make a transmissible genetic "binding" with the One Power (which could happen I guess, some incompatibility between the "binding" and natural life transmission, whereas the TP, not "natural" to life in the Pattern in the first place does a different binding that can be transmitted) , as based on the limited evidence we have, no Construct made by Aes Sedai can reproduce (but however they seem not to have programmed cell degeneration either... Someshta remained alive for +3000 years... so did the Chora trees of the Aiel). It may well be the same for many species of Shadowspawn, i.e. that some specimens that can't reproduce have been around since the War of Shadow. Aginor seems to have found a way, perhaps this could only be done with the TP, to allow Constructs to breed (though not necessarily... it could have been an ethical choice of the Aes Sedai not to introduce new species in the Pattern, to rather make sterile Constructs only, their numbers limited to need and thus kept under control) . I'm guessing Trollocs don't have an extended lifespan either, but it's just a guess.

As for souls, my guess is that it's something I'd call the "spark of life" for lack of something more precise, that calls automatically a suitable soul to the new being. For normal beings, a soul joins at the moment of conception. I guess for constructs it's the equivalent of that "moment of conception" the geneticians reproduced in their laboratories.