Roses, though they may look fragile and seem to require a bit of knowledge to fend off the aphids and funguses, are actually very resilient. If the rosebush was mown down, but not uprooted, it will grow back... it'll take a few years to nurture a noticeable growth, but it will happen. The key to any shrub and rose is keeping the root ball intact. If the rose is over 20 years old, the the rootball is about 10 to 15 feet across, and you're better off than you think.
Protect the remaining cane stump and root ball by providing a chicken wire walling around it... maybe place some colored ribbon upon it to warn landscapers to leave this alone. Mulch lightly and add to it a small amount of Miracle Gro for roses to winterize it. If part of the rose base or rootball is slightly exposed, better to leave it that way... don't over-mulch this area so that air can ciruclate and the sunlight can access any exposed roots. If you have leftover banana peels or eggshells, combine them into the soil as well. All this will help it to overcome shock and feed it, encouraging it to start to grow back in the spring.
As for the rose hips: they tend to ripen and swell if they are pollinated properly. They should be turning bright orange, yellow, deep red, or brown depending on the variety. If they have dried up and fallen off the stems, they will not be of any use to you. They should be remaining on the stem and turning color.
After this preparation, you're ready to germinate them. At this point I should warn you not to expect to produce a rose identical to the original rose, especially if the parent rose is a hybrid. It may revert to some combination of characteristics of its parents.
Select a rather shallow tray or flat with several drainage holes and fill it with fine sand or vermiculite. You can use a Styrofoam tray and poke some holes. If you happen to be near a college with an agricultural department, they will sometimes provide supplies, like flats and soils, for free.
Plant the seeds about 1/2 inch deep. Water thoroughly. Don't flood, though. Keep the soils moist, not soggy, and warm (about 55-60 degrees F). Provide no light for the first month, then give the tray 16 hours of light a day. You can get a 'grow bulb' from Home Depot, Lowes, or maybe even Walmart.
Transplant these seedlings into well-drained, large plastic or metal pans, or flats filled with this mixed soil. Give them a good and full 16 hours of light per day in a warm place (at least 70 degrees F)...perhaps moderately sheltered, but near a radiator, heat vent, or electric baseboard if you're doing this over winter months. A 'grow bulb' will provide this consistent, controlled lighting.
Water sparingly and blot off any water that remains on the leaves. If water is
left on the leaves, sometimes a mildew or fungus can develop, which is bad. Give the
seedlings plenty of air circulation. After the first true leaves form, you can
then transplant them into 3" or 4" pots with the same soil mix, or you can even
move them directly into the garden (after you address and amend the soil by
adding fertilizer or lime). Seedlings may bloom when they are around 2 months