Well first of all you can't store your photos on the camera indefinitely. So you do need somewhere else to store them. If your computer doesn't have much hard drive space, you can get a large USB external or jump drive and use that for storage. I'm assuming you have more than one available USB; if not, you can disconnect something temporarily or get a USB hub.
If you are shooting JPG you have options on file size. I don't know where this is on the D5100 menu but you should be able to find it in your user's manual. You can pick a smaller size if you want to email immediately.
Another option is to keep your larger file sizes and upload them to a site where you can share them. That way you won't have to email them.
working on it in Middle TN Nikon D3100
35 mm 1.8 Nikkor 18-55 mm Nikkor VR 55-200 mm Nikkor VR 55-300 mm Nikkor VR 150-500 mm Sigma OS Feisol CT3471 & Markins M20 ballhead