Upgrade MacBook Pro (2010) vs. save for new computer

I've been rocking a 2010 Macbook Pro, 15 inch, for 3 years now. It's...OK, not great, and I'm wondering whether I should invest some money into it or save up for a new computer.

The 500 GB hard drive has enough space for my stuff, but at 5400 RPM only, is pretty slow. The computer feels a little sluggish when browsing through my large iPhoto library, editing video in Final Cut Pro X, or mixing a large audio project in Logic. The battery lasts about 3.5 hours. It gets a little toasty. I'm frequently tossing the computer into my bag, so the weight is an inconvenience.

My wife has a 2011 13 inch MacBook Air and I'm jealous of its light weight, long battery life, and zippy performance. The 2013 model's insane battery life could really come in handy on long flights. But I'd be out ~$1800 for a 2013 MBA with 8 GB ram and 512 GB SSD (~$1500 refurbished, which I'd likely go for).

Alternatively, I could invest about $400 in a new battery and SSD for the existing MBP. This should give me better performance than a 2013 MBA and a battery boost. I'd still be left with the temperature and weight problems, and there's the chance that a hardware failure lurks around the corner.

Is there some sort of diagnostic/inspection I can perform on my MBP to assess its health?

Any advice from others in a similar position?