When dialing a Saved Contact, it dial 444 & then disconnect automaticaly

To Set speed dial numbers do the following:
1. In Menu mode, select Phonebook.
2. Select a contact to assign.
3. Select More -> Add to speed dial.
4. Select a number (if necessary).
5. Select a speed dial number you want to set.
The contact is saved to the speed dial number.

Note:
• You can call this contact by touching and
holding the assigned speed dial number in
the dialling screen.
• The first 3 speed dial numbers will be
assigned to the top 3 list on the widget toolbar.

press and hold the button down until it dials

Your device has a saved list of important call numbers. The device has been provided auto dial precedence. Reseting the auto or fast dial would prevent dialing soon after pressing talk or call on the phone. Also, your phone subscriber provides the means of last resort to emergency numbers. For example, you may be out of service, however, your phone despite calling anyhwhere provides free emergency contact inspite of not asking for such. Contact the the phoneprovide for alterations.

Voice dialing is enabled by default on the Intensity. All you have to do to use it (assuming you have at least one contact saved) is press the voice dialing button on the upper right side of the phone (near the camera) and on the menu that pops up, select Call (option #1). Then say the contact name you want to call -- for example, "Call Michael." You may have to confirm your selection, depending on your settings, and then the phone will dial the number you have saved for Michael.

I would contact your phone provider. It could be a coinceidence that the dial tone stopped working.
DSL does not require dial tone to work.

1. Enable technician programming mode
• Press "#" - "PROGRAMMING" is displayed
• Dial "20" - "MMC ENABLED" or "MMC DISABLED" is displayed
• Dial the password (default - 1234) - if wrong password is enter "ERROR" is displayed
• Dial "1" to enter program mode
• Press "#"
2. Press "#" key - "PROGRAMMING" is displayed
3. Dial "55" - the display shows "YY MM DD W HH MM" and then the current date is displayed
4. Enter the new date
• YY = Last two digits of the year
• MM = Month (01 - 12)
• DD = Day of the month (01 - 31)
• W = Day of the week (1 -7 day of the week begins on Monday & ends on Sunday)
• HH = Hour (Use 24 hour clock)
• MM = Minutes (0-59)
5. Press "#" - new date is saved

Went into "Contacts" and selected one with speed-dial to edit. Made no changes but saved the contact. Magically, speed-dial began working correctly for all numbers. Rebooting or removing/reinserting the battery may also have worked but were not tried.

Pause seems to work ONLY in saved speed-dial numbers.
BUT...
One work-around to dial any number you want with a pause is
to save the number you want to dial as a speed dial number.
When you enter a speed dial number to save to memory, hitting the "*" button repeatedly cycles through the symbols, one of which is the dash. The dash symbol will be dialed as a 2 second pause when you use that saved speed dial.

Yes, it is pretty stupid that you cannot use this same method (hitting * key to cycle through) to insert a pause in a manually dialed number.
No, I do not know a way to do it in a manually dialed number.

From the phone's user guide:

Set speed dial numbers
1. In Idle mode, press <Menu> → Phonebook →
Speed dial.
2. Scroll to a number you want to set (2-9) and
press the Centre key.
3. Select a contact to assign to the number.
The contact is saved to the speed dial number.

You can now call this contact from Idle mode
by pressing and holding the assigned speed
dial number.

More info about your phone:
http://downloadcenter.samsung.com/content/UM/200802/20080205151101875_SGH-J700_UG_Open_Eng_Rev.1.1_080125.pdf

to store a speed dial on the kx-t7730 with the Advanced Hybrid System KX-TA624 you press the following keys

<pick up the receiver>
2
*
<the number you want to have as speed dial>
<the number you want to dial>
<press the square shaped key next to 0 and 9>
<wait for dial tone to return>
<hang up>

To call using the now saved speed dial, press the square shaped key and then the number you have saved to speed dial

For example

<pick up receiver> 2*1555555555<square key> <wait for dial tone> <hang up>

This will mean that when you press the square key and 1, it will dial 555555555