Here are the answers
1. ITR 3 is required to be filled by you for the F. Yr. 2017-18, for the simple reason you have income and losses from gambling and trading. This ITR is for people who have income from Business and Profession, and gambling, share trading, etc. counted within the Business and Professional Income.
2. Your mother, if she has any income, will need to file ITR 1, provided her income is not above Rs 50 Lakhs in last year.
However, these answers (1 and 2) are valid for F. Yr. 2017-18 (A.Yr. 2018-19). If you have these affair in the current F. Yr. (2018-19), the ITRs will be filed next year after 1 April 2019, and there may be different ITRs then. So depends when you earned or lost money in gambling/trading.
3. Inherited money is not taxable at all. There is NO double taxation of the money already taxed in the hands of your father. However, the income you and your mother are going to earn on the investments using the inherited money will be taxable in your respective hands. There is a legal way to save taxes by way of putting all the money in a separate bank account opened in he name of your HUF (with your mother and you being members and you being Karta) - and then using that money to buy investments/properties further in HUF name. In case you are married, its better to keep your share of the inheritance in the HUF account (you and your wife will be the members and you the Karta). Your mother can continue to hold the investments in her individual name, assuming she does not have any huge income to push the rate of tax to beyond 20-30%. All depends on what are your current state of affairs. HUF is a legal way to make investments in Family (HUF) name so you get all exemptions, deductions, and exemption slabs, in addition to your individual income related exemptions. HUF would need to have a separate PAN number, and you can say the day the HUF was formed was the day of your parent's marriage, and that you are the current Karta after your father passed away. In case of you and your wife form an GUF, the date of your marriage will be considered as the date of formation of the HUF. Find me on Linkedin - amitnarula for more details. I also need to understand which School of Thought you follow for HUF - Dayabhagh (prevalent on West Bengal) or Mitakashara (rest of India). Situation may be different in these two cases.
Disclaimer - These are my personal opinions based on my understanding of law and I cannot be held responsible for any action taken based on the above opinion. Please seek expert advice before taking any action.