Private owners are able to transfer the warranty to a new owner, trade are not.
Yes you can take out a warranty from a dealer after you purchase, the only thing is if the vehicle is out of warranty cover then an inspection may need to be conducted to ensure there are no pre-existing faults before the new policy can be taken out. Some dealers may charge for this, others will waive the fee if they are carrying out a servicing, or if it has full Lexus service history they may just do a quick check without charging. Worth speaking to your local dealer to confirm (service department not sales 😉 )