I'll add more color to Stan's answer (which is absolutely correct, +1), for your consideration.
Irrevocable Trust is a Legal Entity
The trust is an entity of its own, like a corporation or a person. It is not you (the trustee) and it is not your father (the beneficiary). It is defined and governed by the trust document, and you the trustee are bound to act as the trust document says you should act.
Fiduciary Duty
Fiduciary duty is your duty, as the trustee, to act in the best interests of the beneficiary and according to the trust document that created the trust. You cannot distribute moneys to your father that the trust document says you shouldn't, and you can't also withhold moneys from your father that the trust document says should be distributed.
Within the requirements of the trust document, you as the trustee should choose the best alternative to the beneficiary. So if the trust document is silent on the matter - you can decide what's best given the situation, but if the trust document is explicit - you follow the document.
Trusts are Taxed Higher than Individuals
Trusts are taxed at higher rates and allowed less deductions than individuals. So any income retained in the trust and not distributed will be taxed at higher rates than had it been distributed to the beneficiary.
For completion, answers to your questions (which are the same as Stan's):
Trust Distributions Retain Character
Whatever you distribute from the trust retains the character it had in the trust.
Distribution from income not yet taxed - taxable income to the beneficiary in the same category. Dividends - are dividends to the beneficiary. Capital gains - remain capital gains, long or short. Rental income, etc. Deductions from income also flow down to the beneficiary with the income (this is especially relevant to rental income or other types of passive income with high expenses).
Distribution that was taxed before - is not taxable income to the beneficiary. That may be distribution of assets contributed to the trust, or income that the trust retained from prior years and didn't distribute.
Trust distributions and characterization of income are reported to the beneficiaries on form 1041 Schedule K1.
Debtors Cannot Take Over the Trust
Unless the trust explicitly states otherwise, debtors cannot take over the trust - being trust a distinctly separate entity and not your father. This is different from revocable (living) trusts which are not distinct.
What the debtors can do is garnish distributions. So whatever the trust was required to distribute to the beneficiary will go to the debtors directly (but K-1 will still list it as a distribution to the beneficiary).
This can only be done with a court order.
Talk to an Attorney you Trust
I'm not an attorney and the above is not a legal advice, obviously. If you do not trust the attorney you're working with - find a one you could trust and be satisfied with. This is a high risk situation both for you (since as a trustee you may be liable for the breach of the fiduciary duty) and for your father (since his rights as the beneficiary are affected). Make sure you work with a good attorney and clarify and confirm all the above points with them.