I believe the Griffin StudioConnect does not have a mic input, just a 1/4" guitar/bass, line in and MIDI.

For $150, you can get the iRig Pro which has an XLR connector for mic, a 1/4" for guitar/bass and a MIDI interface.

You can plug any xlr mic into it, they are inexpensive, widely available used on craigslist, and come in a huge range of types. The $100 Shure SM58 is well regarded. The Behringer XM8500 is ony $20, and will probably work great for you as well, so you can see the versatility in having a mic preamp and XLR connector like the iRig Pro provides.

It also has 30 pin, lightning and usb cables so you can hook it up to mac/pc/old ipad/new ipad. I have one and like the device, it's very compact and sounds good. I carry it and it's cables in a Tom Bihn Organizer Pouch

For $160 you can get the Presonus AudioBox iTwo which has a second xlr mic/1/4" input, allowing you to record acoustic or electric guitar and vocals simultaneously, for instance. It also comes with mac/windows software to edit your performances.

Then going on up they keep getting more expensive and have more features. I like having the mic preamp in the same device and the ability to hook it up to my mac or pc, so I spent the extra money on the iRig Pro. If I were doing it again today, I would probably buy the Presonus for the larger, sturdier box and the second input.

Have fun!

If you want a dynamic, the Sure SM58 is a good one (generally runs ~$100), or the AudioTechnica Pro 41 is another good one. You will also want to get an interface to control the gain and get a better analogue/digital conversion for the signal than what you can get from your computer. You can go cheap, like with the Behringer UM2, or more expensive with something like an Audient iD4, but if you go cheaper you might need to get a Cloudlifter to get high enough gain without adding hiss from the preamps into the signal.

Depending on what amp you are using I wouldn't do that. The headphone out is already amplified and going to give you a line level signal which is going to be significantly louder than the normal mic/instrument level the amp is expecting.

If you really want to run that specific mic through the amp you would need a reamp box but seeing as they are expensive and you normally don't use a condenser mic on a live amp (they are very sensitive you're probably going to get a lot of feedback) I would just get an sm58 and use that instead.