Cat7 connectors are compatible with Cat5, with enhancements and caveats
Cat5, Cat5e, Cat6 and Cat7 are physically and electrically compatible. It's when you try to do 10Gb/s or higher signaling on the cable that the Cat5e standard falls over.
The physical connectors for Cat6 and Cat7 cables are still RJ45 plugs, but they're shielded and the standard calls for much less untwisting at the ends of the cable in the connectors than what is permissible for Cat5e. It's "harder" to do a proper Cat6 or Cat7 termination.
Aside from the wire twist requirements, in order for shielded connectors to work as designed, the grounding conductor in the cable has to be permanently in contact with the metal shield on the plug or jack when it is connected to the cable. You'll also see shielded Cat5e in installations like outdoor radio connections. Properly shielded cables also open up a whole new realm of potential issues like ground loop interference, so all of your equipment needs to be properly grounded, as well.
Anyhow... if you connect your Cat7 cable to Cat5e jacks, presuming your Cat5e jacks let you jam the thicker Cat7 wires in (which they almost certainly will), you can expect Cat5e performance. Cat5e performance is 1000 Mb/s (Gigabit Ethernet, or about 125 Megabytes per second). I wouldn't expect Cat6 performance out of that configuration (which would be 10 Gb/s, or about 1.2 Gigabytes per second).
Just for reference, if you have a mechanical hard drive in your computer, it's going to struggle to push more than 120 Megabytes per second. Gigabit Ethernet is every bit as fast as your internal hard drive throughput. If you have an SSD, you may see 500 Megabytes per second, which is "faster" than Gigabit Ethernet. With the right switch, you can aggregate ports to create a 2 Gb/s, 3Gb/s, 4Gb/s or faster channel.
Personally, I would run at least 2 Cat7 cables to every room.
10Gbps Ethernet still costs over $100 per port right now, which is expensive for home use (you could get an 8 port 10Gbps switch for about $850).
It's unlikely that you need bigger bandwidth at your house than what Gigabit Ethernet provides today. On the other hand, I'd check the cost of Cat6 or Cat7 jacks and if you can bear the extra cost, have those installed instead of the Cat5e jacks.
Bear in mind that if your installer doesn't terminate those Cat6 or Cat7 jacks correctly (properly bonding the grounding conductor in the cable to the shielding, wire twist rules, etc.), then they aren't really Cat6/7 jacks, anyway. Your installer might know this, and might not want to go to the extra trouble if terminating those cables in full compliance with the standard.
Future of copper Ethernet
By the way, copper network cable has plenty of life ahead of it, out to 50 Gb/s or 100 Gb/s (well over 10 Gigabytes per second), depending on the standard and the cable. Ethernet over optical fiber follows essentially the same roadmap--same speeds--running out for several years, then jumps to bandwidth like 400Gb/s and higher. Who knows if copper will continue to support those kinds of speeds. But are you really worried about being able to move so much data around your home network that 50Gb/s or 100Gb/s couldn't meet your needs? So you can install fiber if you really want to, but you're going to pay a lot more for every port for a zero or negligible benefit if you go that route.
Having said all that, I actually do strongly agree with the notion of running conduit so that you can pull new cable later. Because who knows what's really coming a few years out? One reasonable way to do this is to run the conduits straight up the wall channel from the wall boxes into the attic. You can pull cable straight up into the attic, then run it through the attic suspended from the roof trusses or even strung across the joists.
But in terms of future-proofing, take a look at the Ethernet Roadmap document from the Ethernet Alliance.