Strawberries except for wild cultivars are hybrids and will not come true to seed. Nobody saves hybrid seed. Propagation is usually via runners. For example I planted about 50 plants from runners last year. New plants true to type are not a problem from runners.
On seeds yes they require a chill period. Go to Strawberry.org and you will find all the info you need on strawberries.
If the seed is what you eat, yes cross pollination would change taste. An example is corn, where we harvest and eat the seed. So no your strawberries don't care what pollinates them, though I myself see absolutely no need to hand pollinate. The pollen donor will not change taste of fruit.

Thank you for the good reply, I hand pollinate as i have them under cover.

hmm didn't know the didn't go true. I'll await the runners then.

and good to know i can use any strawberry pollen.

thank you

The wild types come true, like Alpine, and I think Musk too? Again check Strawberry.org, a great resource!