Oncidium urophyllum
 

Is this plant a sequential bloomer? This spike is about over foot long & seems to be finished blooming. But, what are these tiny green things on it?

Tolumnias (which this is... current name is Tolumnia urophylla) will often bloom on the ends of old spikes, or branch and bloom on those branches. So don't cut old spikes until they are brown and crunchy and definitely dead all the way back.

Thank you

Also, it appears to be a climber because it has new growths coming out in a straight line. Currently it is in a
noodle basket. Should I mount it instead?

If your conditions can accommodate a mount (because they need watering more often), it sounds like that would be a "natural" way of managing it, the ideal approach. If not, you can perhaps guide it around its basket so that it stays where it gets moisture.