Q: OS upgrade

If you can't download El Capitan then Lion may be the limits on your machine. If so then there is nothing more you can do.

Unfortunately, v10.7 Lion is an outdated OS X and you may have issues trying to access certain types of content such as a webinar.

Since you had problems downloading El Capitan, try using a different browser such as Firefox for webinars.

You can see El Capitan v10.11 requirements here >  Upgrade to OS X El Capitan - Apple Support

"Do I have to download each of the OS systems in-between OSX 10.7.5 and Captain?"

No.

You do not require an intermediate OSX versions (and in fact some are no longer available) but your computer must meet El Capitan system requirements.  It may or may not be possible to upgrade your computer to the required system. Find your computer model on the http://www.everymac.com web site and near the bottom of the system specification section it will say what OS versions it is capable of running. If you cannot run a newer system you will not be able to sync this phone to your present computer. If it can run 10.8.5 or higher you can either buy a download code for OSX 10.8 Mountain Lion from Apple online, or you can try installing the free El Capitan OSX 10.11.  El Capitan may run slower on older machines and require additional RAM purchase.  Doing a big jump in system versions is also more likely to affect old software.

Snow leopard to el capitan, will it cause my macbook to be slow?  - https://discussions.apple.com/thread/7412959

Mountain Lion 10.8 purchase link U.S.A. - http://www.apple.com/shop/product/D6377Z/A/os-x-mountain-lion

Mountain Lion 10.8 purchase link U.K. - http://www.apple.com/uk/shop/product/D6377ZM/A/os-x-mountain-lion

Current OSX general upgrade information, including system requirements - http://www.apple.com/osx/how-to-upgrade/

Alternatively try a different web browser such as Firefox which is still being updated (for the time being) for 10.7