There are a few problems here, the main one being that young-earth creationists seem to have this picture in their minds of "Mountains" being massive snow-capped mountains much like the Himalayas. This picture from "Answers In Genesis" will demonstrate what they seem to think about a local flood.
The other problem with this question is that "water always seeks its lowest point" isn't exactly accurate. Many floods occur in areas that directly border the ocean/seas, where wind and tidal forces force the water up onto the land. You can see an example of this here:
Well, I certainly do not believe that Noah took anywhere near as many animals as most global-flood theorists believe. There is no evidence from either science or the biblical text that Noah took penguins, polar bears, kangaroos, non-avian dinosaurs, giraffes, etc.
The fact remains that most species of animal would not have enough time to evacuate the area, they would drown before they could reach the edge of the flood zone. Young-earthers often bring up birds, but most species of birds simply cannot fly during torrential rains.
I also feel that it would not leave the impact God intended to bring about if Noah and the animals simply left the area before it flooded. Noah needed to witness the flood and the destruction.
God had also instructed Noah to take more than two of the "clean" animals. This indicates that the animals not only served to help repopulate, but also to continue Noah's agriculture.
Furthermore, there are certain species of animals that are native solely to the Tigris-Euphrates valley (What I think is the best candidate for the flood-zone). These species could have gone completely extinct if Noah had not taken them. It would also take a much longer period of time for the area to replenish if the plants and animals all had to migrate in from the edge of the flood-zone.
3. Why did Noah need to build such a large boat for a mere local flood?
I would say much of the boat would be needed for storage of the vast quantity of food they would require, not just for the eight humans on board, but all the animals on board. Remember, it had to last them more than a year.
Although Noah only took animals from the flood-zone, that is still a large amount of species. An experiment you can do at home, for people who live in the U.S, try to take two of every single species of animal that lives in your particular state. Give them reasonable enclosures (God is not an animal abuser), enough food to last a year, and separate areas for any species that can harm each other. I'm surprised the ark wasn't bigger!
I have a better question for the young-earthers, why did Noah make the ark so small for a great global flood? (I am aware young-earthers attempt to get around this by weirdly equating the biblical "kind" with either a genus or family rather than a species, but this is absurd in many ways that I will address in a future post).
4. God promised he would never judge humanity with a great flood again, if it was local, hasn't God lied? Haven't there been many local floods sense Noah's Flood?
Well for one, no, God has not lied, God never lies, this is a silly question to ask fellow Christians.
God said he would never judge the world like that again (World meaning all of humanity). The flood killed all people except those on the ark. I think the text of Genesis 11 makes it very clear that humanity had not yet spread all over the world, the flood takes place when humanity only inhabited the Mesopotamian region. The fact that its a local flood and thus didn't kill penguins in Antarctica and wooly mammoths in Russia (Which I find frankly silly) is irrelevant to God's promise.
As a side note, young-earthers often bring up the rainbow God used as a sign to Noah and assume that it was the first rainbow ever. I would like to ask young-earth creationist why on earth they get that impression from the text? My guess is, they don't, they just heard it in sunday school when they were kids and assume it to be fact with no objective basis. That is really the general nature of young-earth creationism.
1. Meaning Of "Har": http://biblehub.com/hebrew/2022.htm
2. Flood Video: https://www.youtube.com/watch?v=IAa8BmuPqh8
3. Number Of Species: http://news.discovery.com/earth/plants/874-million-species-on-earth-110823.htm
4. Wildlife Of My Proposed Flood Area: http://en.wikipedia.org/wiki/Wildlife_of_Iraq