Well, when these reputed companies ask you how many days of notice you have, you have two options:
- Tell the truth - "I am available to start immediately!"
- Lie about still being in work and make up some date when you will supposedly be
Now, if you read questions about interviewing on this site, you will find wide agreement that it is a really, really bad idea to lie in a job interview. What do you think is going to happen if you lie and they find out (because people know people) that actually you are no longer working at the company you claim to be working at? That actually you quit months ago?
You've just lost any opportunity to be hired, I'd say.
Now, you may be worried that some employers have a prejudice in favour of hiring people who are currently in work, rather than people who are currently unemployed. This is probably true, it's unfortunate, it can't really be helped. Telling the truth may hurt some of your prospects. But lying is much more risky.
Also, bear in mind that your immediate availability is a positive factor. I don't know what country you're in, but if two month notice periods are standard there, then the fact that you can start immediately rather than in two months time is an even bigger positive. Sell yourself. When they ask what your notice period is, tell them that not only are you available immediately but that you're super excited by the prospect of joining their awesome team immediately!
As for your second question, if one potential employer asks if you have any other job offers? That's a question which, regardless of your circumstances, you should be polite but non-committal about. Tell them that you are talking to various companies but do not say that you do or do not have any firm offers. That's really none of their business.