Can I run my guitar through my laptop for effects and then to my amp?
You certainly can. I own Guitar Rig and I think it sounds very good. It's not free, though... (not free AT ALL)
You can even buy a foot controller to use it as a pedalboard. I have not used this software this way, so I cannot testify to its live usage. I use it to lay down some ideas for future reference. My keyboard player does use a laptop/samples setup and he runs it through a tiny, portable 2-3 channel mixer with EQ before going either direct or his keyboard amp (I'm assuming he does this because he uses two keyboards, but I am not sure). I remember when Mac rolled the new iOS, he did experience some compatibility issues, so if you are going this route, always make sure everything is in the up and up; and that if you don't care for the latest software update, that setup you have allows you to rollback to the previous version.
You would need a powerful laptop and an ADA converter box to plug your guitar into the computer, as well as to go out to the amp. Make sure this box is fully compatible to whatever platform you are using (Mac/PC/other). For instance, Mbox with PC, good. Mbox with Mac, not so much (at least the last time I tried it -years ago-, Avid and Apple were going back and forth about compatibility. That may be different now, but I don't know).
It's a pretty good investment. Make sure this is what you really want. If you live in a big city, you might even be able to try this at a store. I am not even sure if stores do this, though. I am speculating.