Why is Bitcoin called digital gold?
Don't take this too seriously it just a name to praise Bitcoin, Bitcoin is not actually gold. It would be correct if you say Bitcoin is a digital currency.
Does Bitcoin automatically go up and down or is it managed
by any company or person?
Obviously you haven't done any tangible reading, may be you should start. But I will tell you this that Bitcoin is volatile and not managed by any Central body.
If I buy Bitcoin, will it be more profitable to hold it for a long
time or for a short time?
That's up to you because you know your target and if holding Bitcoin short term reaches your target then that is for you to decide but to actually give your portfolio time to grow then long term is great.
As a beginner, what should I do here? Should I slowly start
learning from here or should I try to invest along with learning
This is not a question that needs you to have experience in Bitcoin to answer. It is obviously a generally wise to learn about something first before investing.