When Zoysia is slow to get started for whatever reason then weeds will opportunistically try to get established. You may be able to examine each weed patch to determine if there is a central point from which each is growing, in which case remove the crown at soil level and the problem goes away for a while giving the grass time to re-establish itself.
Now for the grass itself; you can get more information by cutting out a small (in diameter) plug with a trowel or similar digger. Pull out the plug intact - you are looking for the soil and grass profile, that is how much sand or clay is in the soil, whether the Zoysia rhizomes (roots) and stolons (above ground shoots) are present and the depth of thatch (dead grass leaves that accumulate). Because Zoysia is a vigorous grass it can develop a lot of thatch which can prevent healthy spring growth.
Likely you have a free draining soil which can lead to drying out of the grass causing it to lay down combined with deep thatch in places which smothers new growth. The plug profile will tell you this. If necessary consider renting a de-thatcher which is quicker and less messy than trying to do it by hand.