It's been a fair while since I've been there ('89) but I would suggest that the best bet is to leave your bike in Kathmandu and take the bus to Jiri. There are a multitude of guest houses in KTM that are used to storing goods for weeks while trekkers and climbers head out so you should have no problem in finding a place to store your bike.
Jiri has probably grown a fair bit in the last few years, but from there the road ends and the trailhead starts. You could very likely store your bike there if you want but keep in mind that many people will skip Jiri on the way in, out or both by flying into Lukla.
The walk from Jiri to Lukla is about a week, a great way to get in shape and acclimatize on the way in, on the way out it can be a bit of a let down after having been in the heart of the high peaks at base camp and the descent into the foothills (a relative term) will be less dramatic. If you have the time and the inclination, hiking in and out is a great walk if you like hiking, but if you are limited for time you can use that extra week to hike into some of the side valleys when you are higher up such as Gokyo.
Waiting around in Lukla for the weather to clear and the flights to leave can be patience wearing, so we avoided that whole thing by walking out, which didn't seem to be that common. If you do choose to do it, it does go well as you legs are in shape and the air seems thick in comparison to higher up so you can probably get out in five days moving at a decent clip.
There is definitely no riding in the park and not even any mountain biking although back in the day before it was made illegal we did run into a small group that were hike-a-biking their old gull wing Rocky Mountains up to base camp.
It's a special place, have a wonderful trip.