Physical difference between Cat 5 and Cat 5e cabling?
From looking at your pictures, it looks as though you have cat5(ish) wires in the walls and a totally wacky termination at the jack.
You should put a tone generator on each plug and see if it is a single continuous run of wire from jack to jack in your house.
I'd be willing to bet that it is wired in such a way. This sort of topology would work fine for voice but not at all for data.
In such a case, you can probably replace each single "cat5" jacks with a pair of cat5 jacks and terminate each end properly. From there you can connect a switch to each port and get ethernet from one end of the service to the other.
This would be significantly less work than running new wires and give you marginally decent connectivity. It wouldn't be nearly as good as a traditional hub and spoke topology, but it would be much better than wireless...
Good luck!
(allow me to add: In no way is this acceptable work. If I were at a commercial site or if I had just paid someone to do that work, I would tell them to do it over again. Given that this is old work in a residential site, and given that a typical domestic situation's "data" budget is quite a bit smaller than even a small business, I'd be inclined to try to make it work before throwing in the towel and opening up the walls.)
Gigabit ethernet runs fine on cat5, so long as it really is a cat5 installation.
Sometimes 100mb runs fine on something that isn't really cat5, or if you only have 2 pairs hooked up, and then when you put gig devices it fails, but that's because the cables aren't even cat5.
Worst comes to worst, you have to reterminate the ends because the installer didn't terminate them properly (typically by hooking up only 2 pairs or by untwisting 8 inches of the wire or other lazy nonsense.)
Lastly, they haven't made cat5 since 2001 and I haven't seen a cat5 cable that wasn't really old in years. The differences tend to be the number of twists per foot and slightly better quality control on the terminations.
If it is cat5, cat5e, or cat6, or cat6a, the cable will be round and smooth. If it is cat3, it will likely be lumpy and it often kinks or has non-smooth radius bends. If it is something wacky like thermostat wire or similar, there probably won't be 8 wires coming into the jack.
In other words, don't worry about it.
Cat 5 was made obsolete. The Cat 5e specification specifically added some bits that defined specs for crosstalk. In theory, if you have a horizontal run of cable that bumps up against the 90 meter "limit", Cat 5e should be less latent than Cat 5.
In your situation, the difference is nominal. Being that it's a home setup, you probably don't have 100 meters of wire in any one run, so you wouldn't see any noticeable difference. The only "physical" way to determine if you have Cat 5e (if you were REEALLLYYY curious) would be to look at the jacket on the cable. The cable type is usually printed right on the jacket itself.