I'm trying to help him out. His questions are pretty well unswered by mitzimeow, so I thought of rasing a question he might failed to ask himself.
With regards to Nepal, I am yet to meet anyone who has not thoroughly enjoyed their trekking experience.
Perhaps because most people who go there already know they like trekking. It's kind of weird to me if someone never did any hiking until his thirties than wants to start by going for a month long trek.
I loved cycling but only as a means of getting around, never did more than a 60km long trip. If I wanted to start cycle touring I wouldn't start to go for a month even if I'm in a good shape. I'd start by taking several day trips, than say a 3 day long trip, than a week long trip and I'll only start cyling to Africa or Tibet after that.
OK, this is something OP has to decide himself I just raised his attention.
To answer his original question:
Pretty much the only destination where you don't need a tent for trekking is Europe, but October-November is the bad season for that. Most huts are already closed by that time. The other place is Nepal(but only the 3 most popualr regions have tea houses, the rest is again tent only) and October-November is the ideal season. The mountains of New Zealand also have huts and November is spring and it could be a good time(weather my not be ideal, the the crowds are thinner). The problem is the most popular trails have limited access for conservation reasons and slots for trekking permits can get booked out months in advance.