Before faffing about with plastic drain plug on the radiator ---drain the system by removing the rad bottom hose- this is perfectly acceptable practice- in many cars its the only way to do it. Use a proper jubilee clip when replacing.
Its important with these projects to " triage" the car.
It's all too easy with these projects to get sidetracked into non critical areas and spend too much time/money/ resource there. Buy an oil extractor as Pontoneer says- you can use it on your other or future cars.
Its important to focus also on your own safety- you have already mentioned the hazards of working under the car- I would add to these eye protection- working under the car is a messy business and inevitably stuff tends to drop in your eyes- hand protection- disposable gloves I find the blue nitrile ones are best [ You will be already familiar with their benefits in other situations
] and a good set of face masks especially for working on the brakes. But the greatest hazard which may well apply in this type of situation is
fatigue It's when you are tired accidents happen. Its amazing what a little help can do [ even if its just handing you the right spanner or cup of tea] and not trying to rush things.
Good luck with the project. And take stock at regular intervals.