I have a cat 7 network but the router, switch and APs are hooked up to it using cat 5 cables. Does this reduce the advantages of a cat 7 network?
You simply cannot mix shielded and unshielded parts in a link. The shield only works if it continuous end-to-end and properly grounded on both ends. The actual wires in shielded cabling cannot meet specifications without working shielding. Having unshielded connectors means that the shielding is broken on the link, and it is not properly grounded. Improperly grounded links with shielded cable cannot meet the required cable specifications.
There are documents that explain things for you. For example, Shielded and unshielded twisted-pair cable revisited:
If STP cable is combined with improperly shielded connectors, connecting hardware or outlets, or if the foil shield itself is damaged, overall signal quality will be degraded. This, in turn, can result in degraded emission and immunity performance. Therefore, for a shielded cabling system to totally reduce interference, every component within that system must be fully and seamlessly shielded, as well as properly installed and maintained.
An STP cabling system also requires good grounding and earthing practices because of the presence of the shield. An improperly grounded system can be a primary source of emissions and interference. Whether this ground is at one end or both ends of the cable run depends on the frequency at which a given application is running. For high-frequency signals, an STP cabling system must be grounded, at minimum, at both ends of the cable run, and it must be continuous. A shield grounded at only one end is not effective against magnetic-field interference.
Category-7 doesn't actually exist, I think you mean ISO/IEC Class F. ANSI/EIA/TIA define the cable categories, and so far have declined to certify/recognize a Category-7. ISO/IEC define cable classes, and what some people incorrectly call Category-7 is really ISO/IEC Class F. I get suspicious about vendors claiming Category-7 since the specifications and tests for it do not exist. There are far too many cheap cables on the market that do not meet specifications.
Also, Category-5 cable has not been recognized since 1999, and there are no specifications or tests for it.
From ANSI/TIA/EIA 568, Commercial Building Telecommunications Cabling Standard:
Categories 1, 2, 4, and 5 are not recognized as part of the standard and therefore transmission parameters are not listed.
The only recognized categories are 3, 5e, 6, and 6a.