Any work on the lateral line is the responsibility of the homeowner. If necessary, the homeowner should contact a licensed plumber to perform the work.
No, grease should never be poured down the drain. It will solidify in the pipes and can cause a backup in your lateral line. It can also cause problems at pump stations and at the sewage treatment plant. Even if you have a garbage disposal, you should scrape all solid particles of food off your dishes into the trashcan prior to rinsing them in the sink.
The advertising people who decided to label them as “flushable” have likely never worked at a sewage treatment plant. Just because something CAN go down a toilet when you push the handle, does not mean that it SHOULD. Wipes deceivingly labeled as “flushable” do not degrade the same way that toilet paper does and can cause backups in your house, and problems in the collection system or at the sewage treatment plant.
Unless it is toilet paper or something that came out of your body (urine, fecal matter, or any bodily fluid), the answer is no.