When belts wear they lose material or crack. 303 isn't going to help anything but maybe (and that's a strong maybe), make the surface more supple. That's not even considering what contamination of the belt surfaces would do in terms of grip on the pulleys, which could affect operation of belt-driven accessories and create noise. Honda didn't put it on for a reason so keep it off, this is some shade-tree "injineering" alright.
I have the same gen Civic, your R18 engine came with an EPDM belt as stock, not the usual neoprene. EPDM doesn't tend to crack, instead the material is lost between the ribs. This causes the belt to shift slightly, slip or lose grip. It's a better failure mode than cracks and chunks of ribs falling off and EPDM doesn't dry rot as much either. Overall a much longer lasting belt. To accurately measure wear on these types of belts you need a gauge, sight is not good enough.
Gates will send you a free belt gauge. Advance auto parts had a promotion a while back too where they gave out free Dayco belt gauges. On this page click the link that says to request a free gauge. I got mine from them and it works great.http://www.gates.com/products/automotive/tools-and-sales-aids/belt-wear-diagnosis/belt-wear-gauge
I recommend measuring first, because it's free and tells you definitively whether you need to replace or not. You may find it's not worn enough at all and avoid the headache of doing the job.
If you do the job here are some tips:
1. Make sure you compress the hydraulic tensioner slowly, if you go too fast you can snap off the part the wrench or tensioner tool is pushing on and then you're really screwed and will have to replace the whole tensioner.
2. They had a TSB where they changed the belt routing for the 2006 models and beyond due to buzzing caused by the belt routing, TSB 06-059. Whether you already had it done or not, make sure you get the new part number with the longer belt and route it according to the new layout on the TSB.
3. They also had a TSB where 09-007 where the old routing of the belt also put extra pressure on the tensioner pivot bolt which could break it. You may want to replace that bolt like the TSB recommends while you have the belt off, definitely use the new longer belt and routing to put less pressure on the new bolt.
So is brownish color of the belt normal? No cracks or tears or any other visible signs of anything else. Thanks
On the smooth side? Yes. Mine has been brown since the day I bought it, although I imagine this has more to do with their OE belt supplier than anything and not a metric for wear.