Most of the trails in the valley are not very level, but if you walked along the east ridge trail, it'll be mostly flat. Try starting at the skyline gate.
Yes the main trail is pretty flat for a good 2 miles before any incline.
No, Stream Trail is not flat at all, it gradually increases in steepness for a mile before the incline gets very steep, after three miles you are up to almost 1000 ft elevation. I run it almost every Sunday morning and mapped it out on my GPS watch.
Yes parts are level, it depends on how far you go and which trails you take. I wouldn't call any parks in the East Bay hills level. Make sure to pick up a map in the parking lot, its lists elevation and distances.
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