A close-up picture of the flower will help. Regarding the trellis, once a positive ID is obtained, a more precise suggestion on the height of the trellis can be made. For pruning, watch it for one year. See if it is flowering on old growth, new growth or a combination of new & old growth. Then a better plan for pruning can be made. So please provide a close-up picture to determine the ID.

It's already been identified as 'Rouge Cardinal'. This is a pruning group 3 clematis that blooms on new wood. It can be hard pruned to the second or third bud grouping in late winter or early spring. This is not a big vine, maybe reaching 12' or so each season. I don't mind it scrambling on its small trellis and to the fence but you could certainly insert a taller trellis in that position if you wish.

Thank you! How do clematis do "climbing" horizontally, like along this black fence... Will it grow just as large?

With some additional effort, you can make it "climb" horizontally. You should pinch off the tips of new growth and train it along the chain link fence. When you pinch of the tip, at a leaf node, two new tips will start developing from the leaf node. Keep on doing this and you will be able to train it to fill the length of the fence and it will have a much "fuller" look. Don't wait till the buds emerge.