Not going to answer your questions one by one, so just skimming through them:
You will not be able to use any of SharePoint's searches without installing SharePoint. There is no separate search server for SP2013 anymore, it's all one product.
So to answer your question three: SP2013 is better than using Search Server 2010 as it includes some FAST features which you previously had to pay for. For a complete comparison what you get with the free version (foundation) see this page:
SharePoint 2013 feature comparison chart all editions
You can search through any publicly available website with the default SharePoint search, you can also use it to search using webservices or using GET parameters. It would also be possible to directly search through your database using BCS (Business data connectivtiy services), but the foundation version is a bit limited there.
I think the main problem is that you would have to install the whole SharePoint and maintain it as well. I'm not sure it's worth the hassle installing the whole product if you only want to use search. This is exactly Microsoft had inteded for the Search Server 2010 product, but they discontinued it.
Your questions quickly answered:
- Sure, it's a number one product for search. See Garnters analysis about this.
- Search Server 2010, yes you can. SP2013, no.
- 2013 includes the FAST search component, you previously had to pay a lot for. It's better.
My 2 cents: If you only want search, go with a search product like Lucene based products. If you want "more" than just search, or you don't want to get into yet another technology (if you already know some SharePoint) - go with SP.