I have to make a transfer from Alaska to United with only 50 minutes between arrival and departure. What is the fastest way to do this?

There's frequent terminal shuttle from the arrivals level (i.e. lower level). That should beat walking. You still have to reclear security at T7.

The shuttles only stop under the BLUE sign. They won't stop under other color signs. You can ride any LAX shuttle from T3 to T7, whether it says "Airline Connections", "Green Line", or "Lot whatever".

I've seen Lot C buses that were full just go right on by the T7 stop without stopping.

That's not right, unless the driver is sure nobody's getting off.

Are they on separate tickets or is is one booking?

That is a VERY tight connection

Just have to press the call button; don't assume the stop.

Whoever booked that for you did not do a good job. Alaska (OneWorld) and United (Star Alliance) are not partners.

Even if you come in on time this is awfully tight. There is no way to stay in the secure area. And don't count on your bags making it, either.

Explore all options of changing the booking if you can.