The Mishna in Megillah (23b) says they need a Minyan, and the Gemara there clarifies that the groom himself can count as one of the 10. The Gemara (Kallah Rabbati 1:3, Ketubbot 7a) derives this requirement from Boaz (Ruth 4:2).
Although a Minyan is require to recite the full Sheva Berachot, the Shulchan Arukh (EH 62:4) rules that a Minyan is not required to say the blessing "Asher Bara" which is added to all the groom's meals, according to the Shulchan Arukh (:7), and to all of his meals that are not just with family (:7) that have a Zimmun (:4), according to the Rama. The Rama would require a second cup for the one extra Bracha as well (:9).
I'll note because you thought to connect the two, that the special Zimmun for a wedding ("SheHaSimcha BiM'ono") is recited at any meal from the time they begin preparing for the wedding until 30 days after the wedding, and also at any meal made specifically to celebrate the marriage within the first year of marriage (:13). It is not connected to Sheva Berachot at all. That said, the longstanding custom is to only recite it during the first 7 days of marriage for a first marriage, and during the first 3 days for a second marriage, when the obligation to be happy is especially strong (ibid.).
(Some communities customarily recite a poem before the Zimmun, such as "Nodeh LeShimcha" or "Davay Haser". The Beit Shemuel (sk 11) thinks that the latter should be used if all 7 blessings are being recited, but if only the one is being said, then the former poem is to be preferred.)
If you have 7 men at this meal, and you presumably have some guests, then you would say the special Zimmun and the blessing "Asher Bara" afterwards and be thankful for the opportunity to celebrate with the bride and groom in an intimate setting. Enjoy :)