Hi Bigeater
I think the weight issue depends on how active you have been in the past. I started at 150KG and certainly push-ups were OK for me but I played a lot of rugby when I was younger and still carried a fair bit of muscle (under the flab). Planking has taken some time for me to get used to. I started on 5-10 seconds and after six months can now do five times 60 seconds.
For push-ups you could begin by starting on your knees or by leaning on the stairs to reduce the angle and just take it easy. Do the exercises slowly to build up the muscle mass.
Nice and easy and build up slowly. Good luck.