I'm not a successor trustee, but I do help my Dad out with the taxes on a trust related to my Mom who passed away in 2016, so I think I can answer some of your questions.
Yes, the trust will need to get an EIN. You just ask the IRS for one; it isn't hard. You give them the name and address of the trust, and possibly the trustee's name, and they give you a string of digits. That string of digits goes on the trust tax return every year.
Yes, the trustee will have to file Form 1041. I think it's a complicated form, so if I were doing it I would hire a CPA to do it one year then have them show me how they did it. Then the remaining years I would probably do it myself.
The things I've learned about trusts and taxes:
1. Trusts pay income taxes to the federal and state governments on their income, just like individuals.
2. There are income tax brackets for trusts, just like there are for individuals.
3. But the income tax brackets are much lower for trusts than they are for individuals at the federal level. For example, a trust like my Mom's which earns more than about $9,550 in 2018 is in the 24% tax bracket. My conclusion, supported by the CPA who does my Mom's trust tax return, is that trusts are not very good for sheltering income from income taxes.
4. Trusts get a "standard deduction" just like people do, but it's much smaller...like $100 or $300 depending on the kind of trust.
You can distribute income from the trust to the beneficiaries in accordance with the trust's instructions. Any income that is distributed is shifted from the trust's income tax returns to the individuals' tax returns. It also generally retains it's income tax character, so if the trust received dividends on some stocks it owns and distributes that, then it ends up on the recipients' tax returns as dividend income. See Form 1041 line 18 and Schedule B lines 10 and 11.
You're right, the distributions are reported on K-1's. In the case of my Mom's trust, my Dad is currently the sole beneficiary, so we have the same CPA firm do the trust taxes, the K-1, and my Dad's taxes.
What we do every year is I prepare a mock tax return for the trust and for my Dad and try to figure out what income tax bracket each of them is in. If the trust is in a higher bracket, then we just distribute enough money to my Dad so that the higher tax bracket is avoided. This year the two were in the same tax bracket so we didn't need to do anything.
Another couple of things that I'm 99% sure are true: State taxes paid by the trust, tax preparation fees for the trust, and legal fees for the trust are deductible against trust income without any sort of "2% of AGI" floor like it is for personal. So keep track of those fees. We pay them from the trust just to make sure that we can deduct them. See Form 1041 lines 11, 12, and 14.
Hope that helps.