Yes, you can do this. The lump sum or payments can go directly in the IRA. Remember that if you get your check sent to you and you put the money in the IRA, you will pay taxes on this. You want the check to go directly to the bank.
This is generally difficult legally. However, you can set up a money market account with your IRA holding company. Then have an amount equal to your pension payment pulled from the money market account and put in your IRA account each month.
No. Contributions to an IRA have to be from earned income. Stock dividends, interest, annuities, capital gains, and retirement income such as social security or pension payments do not count as earned income. Check with the IRS to be sure.
Yes, you can roll over you pension payments to your IRA account. However it does have its advantages and disadvantages too. And the best part is it is tax free!