USB devices take two tries to work, and won't boot with Time Machine drive attached

I have a mid 2015 Macbook Pro 15" from work (so I can't afford to take this into service for days at a time). My specs are as follows:

I am running the latest version of El Capitan, 10.11.4.

My problem is that USB devices--such as external hard drives, Logitech USB headsets, thumb drives, etc.--always, and I mean always, require two plugins to be seen by the system. It never works the first time. I always have to plug in once, remove it, and plug it in again.

In addition, if I try to boot the system with my Time Machine/external USB HD drive, it refuses to boot at all. Only if I reboot it without any devices connected does it boot successfully.

My questions are: is this a software or a hardware problem? And has anyone seen this before? I'm thinking it's a hardware problem, though the latest patch notes imply that the last OS update may have addressed a similar issue with audio devices. (This would affect my Logitech USB headset presumably, but it hasn't helped.) If it's a hardware issue that requires a motherboard fix, I may not bother, since again it's a work machine that I can't part with for a long period of time unless IT is willing to swap me a machine on the spot. It doesn't affect most other day to day operation except I have to plug in devices twice all the time.

Thanks, everyone.