How to restring a Honda gx25 weedeater 4 stroke

If you have proper timing, compression, gas and a spark, the engine should fire/run.
Have you tried a spritz of starter fluid to see if it fires ?
The flywheel could have sheared the key and affected timing.
About the last thing I can think of is valve adjustment, specifically the intake valve, although I think it would be difficult to pull the crank rope if that's the problem.

I suggest you buy the factory manual.

The engine on that trimmer will be a 2 stroke 30cc engine not a 4 stroke.
"Two Stroke Oil" is added to the fuel for a 2 stroke fuel mix in accordance with the proportions specified by Ryobi for that engine. The required petrol/oil mix may be printed on the top of the fuel cap. No other oil is used in a 2 stroke engine.

There are a number of videos on Youtube people have uploaded which will show you how to replace the pull cord on a trimmer.

Okay. You have eliminated the plug. Now disconnect the coil kill wire and try for spark again. Also check grounding between the coil laminates and the cylinder with you ohm meter should read zero.

If grounding okay and with the kill wire disconnected you still get no sparks then replace the coil pack as it probably a failed internal trigger which can not tested for.

This WEB site shows how to replace the nylon line.
http://www.youtube.com/watch?v=to_DQP-XY0M

To remove the head to restring, push in the bumper cap/side tabs on the strimmer head, the top should then just pull off exposing the inner spool.

I also have Mantis tiller for 2 years. From the very beginning I had a problems. I was referred by Mantis to the repair store and was charged $90.00 for oil drainage. The second time I was waiting for handle replacement for ~ 1 month. This tiller can't handle even little jobs and the warranty doesn't worth a dime. Maybe I will be more successful using it in my living room . I have an equipment with Briggs and Stratton engines and I never had a problem.

To me Mantis mean big advertisement and waste of money.

Honda GX25 GX35 manual on PDF.
http://www.smallenginewarehouse.com/images/manuals/GX25-35.pdf

The burning oil for a few minuets is more than likley the valve seals.. They often used plastic ones..

As far as the major leaks - that has to be seals and/or gaskits..
You would have to see exactly whre the leaks(s) is starting from..

jj505 has good and bad information.
Honda GX25 is a 4 stroke motor, with oil lubricated crankcase.
Oil drains out the filler.
Oil and gas do not mix on this four stroke.
The PDF below is the Honda manual.
http://www.smallenginewarehouse.com/images/manuals/GX25-35.pdf