Yes. If you were getting a long action cartridge I would have no real preference beyond personal whim. While the action length difference between long and short action is not huge, pick up one of each and you will see that at least in feeling and handling it is significant.

The other issue is if you may want to do anything to upgrade the rifle in the future. There are innumberable options to upgrade all components on a Rem700 action and a very good number for the Savage action as well. You would have a more difficult time upgrading the Tikka.

Both the Savage and Rem700 are good actions. I give the Rem the edge on the trigger (though some people love the Savage Accutrigger). I have a Rem700 long action and Savage short action with pre-Accutrigger. For an extra $70 I upgraded the Savage trigger (Rifle Basix) and it is now superior to the Remington.

The one interesting thing about the Savage is that if you know what you are doing you can switch barrels (and calibers) all by yourself with minimal equipment. My Savage is set up to switch back and forth between 2 barrels.

I have a slight preference toward the Remington action, though that will be at the very upper end of your budget and the Savage will come in lower for an equivalent rifle.

As for the scope, good bang for your buck is the Nikon Buckmaster. Out the door for around $300 and leave you cash for base and rings. I am a fan of Burris Signature rings. I own two of these scopes and they are a great value for the money.