For all his years in the temple with his mother and other priests and priestesses of Mara, Trygve feels that marriage is something that other people do. Even when he was young, he had a hard time picturing himself settling down with one person. He wanted to travel and explore. With respect to marriage in general, he is more pragmatic than romantic, being of opinion that love is not the only valid reasons to get married, and that marriages that are economically or politically advantageous are equally as valid. However, he thinks that these should be arranged between the participants, not their families when they are young.
As far as children are concerned, he likes them well enough although, as with marriage, parenthood seems to be something that other people do. At the most he could picture himself adopting an older child when his days of protecting fugitive Altmer and wee Dragonbos are over. On a somewhat related note, when became Thane, he tried to convince Jarl Laila to do something about Grelod the Kind and her restrictions on adoption. (His father got sick before he saw this particular project through.)
Trygve would be a fairly strict parent, but not overbearing. Their days would be regimented, organized around chores, studying, and training. He would expect his kids to learn as many useful skills as possible, in addition to learning about Skyrim’s history and religion. Before knowing Elspeth he would probably discourage any magic that wasn’t in the Restoration school (like a typical Nord). Having known Elspeth and spent some time at the College, he’s become a bit more open minded on the matter, though he would continue to discourage study in the schools of conjuration and illusion (and forbid those spells from being cast in the house).