Is it possible to have trekking in the Alps in September and October? It will be warm enough to do such type of activity? Is there high probability of rain and clouds?
You can have warm days and hiking is definitely possible. But the first week of September is very different from the end of October and anything can happen, even snow. There is certainly a decent chance of wet and relatively cold weather but that's not the end of the world, on a multi-day mountain trek, you have to be prepared for that anyway.
The main issue is not simply rain and clouds but very bad weather (which is also possible in the summer, albeit less likely). If there is a storm, snow or foggy conditions, you don't want to be too high in the mountains, especially without equipment and a guide or significant experience. Depending on altitude, snow is definitely possible in October. Be ready to change your itinerary as needed, do not always expect to be able to do everything you planned and do not take risks because you only have a few days left.
Incidentally, I have had near-freezing weather at 3000 m a month ago, in the middle of summer, and that wasn't by any means a particularly bad day or record low (in fact heat was the main problem this summer, causing accidents in the Mont-Blanc area because of melting ice). Whenever you are contemplating a mountain hike, you always need to track the local weather and be ready for sudden changes.
When I was a student in Paris, a girl once asked me to accompany her to Chamonix to go hiking. It was mid-October. I figured it would be cold in the Alps, but when we got there, Chamonix was covered in snow. The girl was determined to go hiking, so we did. Did I mention the snow? There was lots of it.
The next day, we went to St. Gervais-les-Bains for some reason that escapes me. It was raining at the time, but she wanted to see whatever it was. Supposedly it was within walking distance, so we walked from the train station up the road for a few kilometers, getting soaked in the process. We never made it. She finally gave up and we turned around. Some kind-hearted local offered us a ride on the way back. Did I mention the rain? There was lots of it.
Basically, expect snow and rain in October.