You can send videos to recipients from an iPhone 5S, depending on their length. If the video files are large, it may be best to upload them to a cloud service like OneDrive and then share the link to them.
If you are having trouble getting them to your computer, you may have a driver issue. Did you download the newest iTunes for Windows? Otherwise you could use a app for your iPhone called File Explorer. It has a free and paid option. The free allows you to connect to only one external source at a time, which it sounds like is all you would need.
Using that, you could find a tutorial online about how to set up a file share on your computer, then connect to that share with File Explorer and transfer files between your computer and your phone with ease. I did it all the time back when I was running Windows on my computers and now still do it even though Im running Kubuntu Linux on them.
Photos app for Windows 8 or later
Step 1. Connect your iPhone to computer > Open Photos app.
Step 2. Click the device you want to import from > Select the photos you want to transfer, or click Select all: AutoPlay
Step 1. Connect your iPhone to computer, the AutoPlay will appear automatically.
Step 2. Click on Import pictures and videos or Open device to view files.
If you click on "Import pictures and videos", AutoPlay will import all camera roll photos and videos. If you choose "Open device to view files", you need to find the videos that saved in different folders.