1. What color and type/brand of flooring would you recommend? Looking for a "wood look" flooring that will stand up to big dogs, dripping kids and is budget friendly and scratch resistant.

No wood flooring will every hold up to big dogs. You'd be better off going with a wood look tile. They're not the cheapest, but they hold up well and won't need replacing after a few wet dogs run thru your new cabin.

2. What type and color of counter tops? Should the island and counter tops be the same or different colors? I love the look of quart, but would be interested in more affordable options that would hold up well.

Since you say you're going for cozy, I'd suggest using the same color for a more harmonious space. Changing the color works well in larger spaces in my opinion.

3. The rectangular game table will be placed near the entry door. Looking for furniture placement ideas and decorating ideas. Not sure about this one, sorry.

4. What should I do on the wall with the fireplace. My husband is hanging a television at the top...ugh!

Between the windows? The backlighting is going to be a pain to look at while watching dark shows. Please also be careful about curtains at those windows so close to the fireplace.

5. What color appliances?

If budget allows, you can get "panel-ready" appliances and you can put in wood paneling. That could be great for a cozy cabin look. If budget is an issue, then I'd go with stainless steel, should match well with a rustic cabin feel.