This is, in fact, a very common transaction called a Roth Conversion.
And to answer your 2nd question first: No, conversions and contributions are completely separate. They're different things. Contributions involve moving money not currently in a tax-favored account into one.
Your 1st question is more involved... and must be done correctly to avoid going astray of IRS rules. Also, unless you're converting a small amount that can be overcome with deductions you will be paying income tax on any conversion from a Traditional (or Rollover... little functional difference) to a Roth IRA.
A Roth Conversion consists of 2 parts:
1-- Transfer to the Roth IRA, and
2-- Reporting Roth Income on your taxes
You'll have to direct your financial institution or bank-- whomever is the custodian of your Traditional IRA-- to transfer the Traditional IRA assets you want to transfer to an account set up to hold Roth IRA assets (your Roth IRA that's already set up, in this case). I use the term "assets" because you can transfer one of 2 ways: 1-- Cash, or 2) In-kind. This is referred to as a Roth IRA Conversion
An In-kind transfer means you will transfer actual securities positions, e.g. I have 500 shares of AAPL and 100 shares of the IVV etf. On my conversion paperwork I select "In-kind" and designate 300 shares of AAPL and 75 shares of IVV.
This is the simple part of the transaction. Your IRA custodian will most likely have forms formatted the way they want ready for you to just fill out. These forms might even (most likely will) include additional form fields used to calculate the required reporting described in the next section.
The potentially more complicated part of the transaction will be reporting Roth Income to the IRS. This really only has the potential to be complicated in a few cases: 1) if your IRA custodian doesn't have a comprehensive Roth IRA Conversion form that will handle all aspects of the transaction, including the accounting needed for proper reporting, and 2) if your Traditional IRA includes non-deductible contributions from prior years.
If your Traditional IRA consists of 100% deductible contributions, i.e. all of the contributions were made while participating in an employer plan or that didn't allow for or in which you made 0% non-deductible contributions, or an IRA that in which you made no non-deductible contributions, then 100% of the value converted to Roth assets will be reported to the IRS as current income. Your basis in these assets transferred to the Roth will be zero, i.e. 100% taxable.
This is what I referred to above as Roth Income. This amount will be added to your income for the year in which the conversion is made.
You'll receive copies of the forms your custodian files with the IRS. These forms plus copies of the paperwork you used to make the transfer will be needed to properly calculate and attribute income from the transaction when you file your taxes, so keep a file on the transaction to make this a simple, painless process.
The only additional considerations I would add to your rationale for converting Traditional IRA assets to Roth assets are:
1-- The Tax Ramifications of the Roth conversion, and
2-- The Investing Ramifications of the transaction
This is simply a matter of understanding what cash/liquidity you'll need to have on-hand to pay any additional tax due (over and above what you were expecting) as a result of the conversion.
Further, be aware that the Roth Income might be large enough to bump you into a higher tax bracket. If that's the case, you can manage that with your accountant through either offsetting deductions that bring you back to the originally assumed tax bracket or by considering this issue prior to any conversion and adjusting the amount of the conversion accordingly. You can also convert your rollover IRA over several years, too, if this is an important issue for you. Of course, you can always just bite the bullet if you want the total amount converted immediately, for whatever reason (simplicity, investment expectations, etc.).
Last, it's important to factor your belief or assumptions as to future tax rates applicable to you, e.g. if you believe that you'll have to pay a higher % of your income to the IRS when you either have to (due to RMDs) or want to start taking distributions from your IRA and/or rates will go up on your income in the near future, justifying paying tax on the full amount today versus breaking it up over several years.
Understand that when you convert funds from Traditional to Roth assets you'll be paying taxes on the amount converted at regular income tax rates. You'd be wise to consider the effect of this payment on your future returns, based on your age.
This becomes a more important consideration the older you get. That's simply due to the time you have left to overcome the taxes paid on the conversion and realize the additional asset accumulation/accrual benefits of the Roth IRA. This can be expressed as the number of years you'll have to remain invested before "getting back to even" on the conversion.
While you might leave 100% of the assets converted in your Roth IRA, your actual returns are a function of your total wealth* and assumed current and future tax liabilities, i.e. taking into account the $s paid in tax directly related to the conversion vs paying the taxes later in a Traditional IRA. If I pay 25% tax on a $100,000 conversion I'd look at what returns over how many years will be needed until I've recouped the original -($25,000) expense and my Roth is now apples-to-apples with my former IRA (in terms of the benefits/usefulness related to my assumed future needs). This is the point when the Roth really starts to shine due to the additional benefits of tax free growth, the ability to contribute after age 70 1/2, and the absence of any RMDs (Required Minimum Distributions).
But I don't think your question went into this level of depth, so it's best I stop here before writing a book.