There are a few options for the treatment of warts, not all are painful.
You should consult with your OB/GYN for the option that best suits you.

This is variable and depends on the kind of anesthesia you receive. This is a question better served for the OBGYN who will be doing the procedure, knows which technique they will use, and is aware of the anesthesia type being used.

Freezing can be done out of the vagina with freezing, but not in the vagina.

There will always be a small amount of discomfort associated with cryotherapy. Fortunately it is short-lived.

Vaginal warts are removed by various methods. Can use surgical (i.e. laser, freezing (cryoablation)), or topical medication (i.e. Trichloroacetic acid, Podophyllotoxin). Most of these will cause some pain or discomfort in the area, and might benefit from topical lidocaine. Please discuss what might be best for you with your gyn as it depends on size, amount and location.

Yes

The out side on the skin is going to cause discomfort more than the vaginal and when the freezing reaches the base of the wart.but it can be done as out patient without anesthesia

The procedures for removing genital warts are performed under anesthesia, so procedural pain is typically not an issue. Pain after the procedure is managed with both oral and topical medications, but is usually minimal.
There are multiple ways to treat genital warts which do not always involve procedures. You should see your gynecologist to see if you are a candidate for topical therapy with one of the antiviral creams.

Any procedure to remove, destruct genital warts can be painful. Different people have different pain thresholds and handle the treatments differently. Most people do well with the treatment in the office.

Usually the freezing procedure is not too bad.

Sometimes the procedure can be painful, I recommend that you discuss this with your gynecologist at your next visit.

The procedure itself isn't painful, but the post-operative period (while you're healing) can get uncomfortable. Make sure to wear loose fitting clothes, keep the area clean and dry, and always call your doctor if you notice signs of an infection (redness, swelling, drainage or disproportionate pain).

Yes, freezing procedure to the skin are painful. There are different ways to numb the skin before a procedure.

Inside the vagina is more painful than outside

Yes, the procedure can be painful. If there are multiple warts to be treated, getting a local anesthetic first helps a lot.