I have a standard light switch that controls one overhead light - fine. Now I want to replace it with a Leviton T5225 which I think is the right weapon for the job -- one unit that has a light switch and an electrical outlet. I want the switch to control the light and I want the outlet to always be on. Inside the electrical box there is one black wire, one white wire, one ground -- that's it. Can this be done, or do I need a different switch for the job? I'm definitely doing something wrong.

https://www.leviton.com/en/products/6672-1lw

Specs say Dimmable LED lights are fine, so that means your lights are the ones at fault. Although marked dimmable, they are aren't. Swap the bulbs out for another make. Also if you read the FAQ on the lower section of the page, there are secret switches hidden behind the slider. Also interesting to me is that on the top metal fixing plate it shows
LIGHTED OPTION
ON - OFF

This site can give you the support (oper. man. and spec sheet) youi need to correct this problem.Electrical devices lighting controls network solutions Leviton com

1) Guessing you have 3-way switch where 2 different switches control same Load (light, fan, motor). Unknown wiring, and unknown if fan and light are wired separately.

2) Google search for 3-way fan switch shows Leviton IPF05 with simple straight-forward wiring. Link below has instruction manual, and Live chat button located to right of LEVITON
http://www.leviton.com/OA_HTML/ProductDetail.jsp?partnumber=IPF05-1LX&section=47082&minisite=10251

3) It appears that the Leviton device must replace Line side switch.
http://waterheatertimer.org/images/GE-15312-3-way-circuit-600.jpg

4) Uncertain if this device also controls fan since the wiring is so simple.

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

It depends. Some dimmers are considered "smart dimmers," where the control on the other end of the 3-way has to be intelligent enough to send a signal back to the dimmer. If the Lightolier dimmer was a smart dimmer, then chances are that the 3-way control was designed to talk only to that specific dimmer. The Leviton product you replaced it with is a simple, basic dimmer - it should work in conjunction with a simple, basic 3-way switch.

If the 3-way control already IS a simple, basic 3-way switch - then I would simply re-check the wiring of the dimmer you replaced.

I think the matching remote totally matches down to the LED display of light level. Coordinating remote does not have the LEDs.

Connect the white on the switch to the white in your house, one black to the hot coming in, and the other black to the hot wire to the overhead fixture.

To run an outlet you need a neutral, which is not present. I would advise contacting an electrician.

http://www.leviton.com/OA_HTML/ibeCCtpItmDspRte.jsp?item=4044§ion=10103&minisite=10025
The Leviton page for Leviton 5613 does not have a troubleshoot.
Manual shows simple wiring diagram.
3-way switches will not operate lights if switches are wired incorrectly.
Check your ground wire is connected.
Leviton 5613 series is illuminated.
Check that both switches have 5613 number.
Swap switches and see if problem is inside switch.

The Leviton 6696 Anywhere RF(Radio Frequency) lighting control allows you to add an additional (wireless) switch to an existing switched light(s) without having to hard wire it. For example, let's say that you have a bedroom with a switched overhead light. The Leviton 6696 RF receiver would be installed in place of the existing light switch. It will still function and is wired the same as an ordinary switch. The Leviton 6696 Anywhere RF transmitter (after installing the batteries) can then be used anywhere within 50 feet of the receiver (that replaced the existing light switch) to also control the light. You don't have to mount the transmitter if you don't want too. It can be on the night stand, (or carry it with you if you want) so you can turn the light ON and OFF from the bed or wherever.

I recently purchased a Leviton Dimmer switch which has a brown lead. This switch is to replace 1 of 2, 3 way switches that control the ceiling light in my kitchen. My problem is the new dimmer switch has 4 leads including a brown one which has me confused. I have installed and replaced numerous dimmer switches without difficulty. Please help.