Just slap some pipes together! (Just kidding - don't do that!) Based on your "new DIY" posting, I'd strongly suggest getting some help to learn. Even hooking up a plain old sink for a rookie can be a challenge - this is tricky in several respects for a novice. (too many places to have leaks, for one thing).

Assuming that the peninsula is permanently installed running the lines under the cabinet would afford the lines the same level of protection as if they were run through a chase and surely more protection than if it was run through the cabinets, however the actual distance you could run would be dependant upon how the cabinets are made. Typically the vertical clearance in the underside of the cabinet base is the same as the vertical height of the toe kick, which is generally 3-1/2" Under the IRC a kitchen sink drain line is 1-1/2" pipe and the outside diameter of 1-1/2" sched.40 PVC is 1-7/8. Subtracting 1-7/8" from 3-1/2" leaves us 1-5/8" for pipe pitch, (assuming of course that the downstream end of the pipe is flat with the floor.) Code requires a 1/4" per/ft pitch so the absolute maximum horizontal length we could fit in the space is 6=1/2'of pipe, however that is assuming that on the downstream end of the pipe you dig into the concrete and set a Tee so that the pipe will be flush with the floor on the downstream end and that you make the hole in the cabinet base on the upstream end large enough to allow the pipe to be flush with the underside of the cabinet base, otherwise you will loose about 1/4" to the sidewall of the 1/4 bend turning up, which would be equal to 1' of run. The problem is venting the trap. Under the IRC for a fixture arm the bottom of the pipe at the trap weir may not be higher than the top of the pipe at the vent opening therefore you would need to provide a vent near the sink. The IRC allows an unlimited number of AAV's (Air Admittance Valves commonly called Studor vents or cheater vents) You could then connect a vertical riser from the 1/4bend at the cabinet base and run a vertical up to the fixture arm height where you install a Tee, then extend the vertical riser at least 6" and install an AAV. The preferred option would be to install an island loop vent in the cabinet at the sink location. Under the UPC they require the kitchen drain line to be 2" pipe therefore you could configure in about the same manner but the maximum horizontal length that would fit under the cabinet is about 4'.

Some questions... How extensive is your remodel? Are you "gutting" the kitchen? Down to the studs? How flexible are you on moving/changing walls/doors/windows? Is the hand-drawn pic your current kitchen? I'm assuming the dining area is to the left of the new peninsula shown in the first pic (the computer-generated one). Is that your only DR? Or, is it more of a "Nook" for every-day eating and a formal DR elsewhere? If so, where is the DR? Do you have a basement under the kitchen? Crawl space? On a slab? I wholeheartedly agree with Bahacca! Do not plan a kitchen around young children. They grow so quickly (I know *sigh*)! You will spend far more time with them as children/teens old enough to not only not have to be monitored constantly, but also old enough to start helping around the house and kitchen. They will be learning to cook and cleanup and then, maybe, taking on some of these tasks themselves. So, do your best to plan for a kitchen where 2, 3, or 4 people are working...prepping, cooking, cleaning up...all at one time! BTW...think "work zones" instead of just "work triangle". There are three primary work zones: Prep Zone...where 70% (or more) of the time is spent while working in the kitchen [sink (prep or cleanup), trash, workspace, located next to or across from Cooking Zone] Cooking Zone...where 10% of the time is spent while working in the kitchen [range/cooktop, trash, workspace and emergency landing space, located next to or across from Prep Zone] Cleanup Zone...where 20% of the time is spent while working in the kitchen [sink, DW, trash somewhere nearby, separated from Prep & Cooking Zones - even if only by a sink] There's an interesting discussion about work triangles and work zones in the thread I've linked below. Here is a link that might be useful: Thread: Death to the Work Triangle??

Knickema, I've been in your shoes as a homeowner pulling the permit. In my small jurisdiction, the lead inspector told me we needed to do a homeowner/GC permit. I spoke to him before we started planning, and he said that the only way to get the licensed/permitted pros to pull the permit was to have them act as GC for the entire remodel (he recommended against this). So we got the permit. That said, listen to the advice above and find a plumber who is familiar with current codes for island vents.