2000 Ford KA cold start

About five weeks ago, it had rained in morning. When I sat in my opel astra 2000, upon starting all the lights came on but the car wont start. Not even a click.

Then later in the day about 7 hours later, I sat in the car,, turned on the key and the car started normal. About two weeks after, it did the same and I went to check the battery.We put the car to 2nd and a push made it start. It was down, so I bought a new one and replace it. A week later, it started normal, i drove and parked it and when starting again, lights come up fine but the car wont start although I had bought a new battery. In a 2nd gear and push, the car started again.

Then a week, it had rained with humid temperature, it happened again. 2nd gear and a push and the car started. I took to a shop to test the alternator/generator and the results indicated it was good.

Could it be the starter or electrical problem? Sometimes, when I start the car, the sound goes very very high and it then cools down itself. Is it normal?

Now what could be the problem? Any information will be highly appreciated.

Car: Opel ASTRA 2000