We have done a lot of hiking with our baby, but not trail running.
While I think you could, in theory, run with a backpack, it's likely to mean a lot of jiggling for the baby. They aren't designed to be so restrictive - the shoulder straps have a few cm gap above baby's shoulders for comfort and mobility, which would mean a few cm jump each time you take a step. Whether it's safe or not, I would recommend asking your pediatrician.
The baby might not let your partner get very far anyway - I'm pretty sure our baby would find the first couple minutes of running fun, but then would start complaining.
Some info from a blog about trail running:
It is true that every day, my daughter and I were out exploring nature with her in a pack. But run…well, honestly, I would run every once and a while, for a very short distance, just to make her laugh. She would love the rhythm and bounce of it. But can I recommend truly going for a run with a child in a backpack? No I can’t. I think it would stress not only the carrier’s system, but also expose the child to danger. Carrying a 20-50 pounds of anything makes it more difficult to pick up your feet, and tripping while running becomes a real issue. Tripping on your own, so what, you scrape your knee and maybe bloody your hands…but tripping when you have a child on the front or back of you??? Disaster...as well as irresponsible.
Falling is a concern no matter what the speed - but I think it's more likely if you are walking very fast or running.
A baby wrap would be a better choice, since the baby is really attached to you rather than to the straps of the pack.
In response to your question about age limits - the manufacturers give guidelines, but we found that the recommended ages overlap a lot and it comes down to comfort (for you and the baby). For example, we started hiking with a sling (moby wrap) when our baby was 2 weeks old, transitioned to a front pack (ergobaby) at 4 months, and now we use an Osprey paco baby backpack. The ergobaby still works for short trips, but on long hikes it pulls your shoulders too much.
The backpack won't be of use until your baby's spine is strong enough to support his/her head in motion, when s/he is sitting very well independently. We tried it at about 7 months, and it wasn't so comfortable for her - again at 12 months it was fine, now at 15 months she loves her backpack so much that we have to hide it or she'll want to get in it immediately!
Whatever you choose, I'm sure your baby (and you and your partner) will benefit from being in the great outdoors.