If someone wants to buy digital gold or wants to invest in a commodity mutual fund which invests in gold, for short term or long term, until he makes some good profit, then can he do that? Is it permissible in Islam? Does he have to pay Zakat too for his digital investment?
There is no such thing as digital gold. Perhaps you mean bitcoin and its ilk?
At present, most regulatory agencies explicitly advise against digital tender. Moreover, investments are considered haram when the investment is itself haram, and this will depend on a number of factors. For example, investing directly in fossil fuels, given the scientific consensus on anthropic climate change, is haram. Investing indirectly, say through, a fund that spreads its portfolio through a set of different markets, will be haram when the portfolio is predominantly investing in industries that are haram; say for example, in fossil fuels, or in arms firms that have no ethical compunction in supply mercenary militias, or say, in vulture funds that buy up odious debt and demand repayment.