Op If i'm brutally honest she sounds overtired to me. Typically, kids that age are still having a decent 2hr nap, some still have 2 naps & most kids dont drop the nap completely til nearer 3yrs old. I would stop cutting her nap to just 30mins its not enough
If she was only having an hour anyway it may be the timing of the nap was already off so it may be worth posting what her daytime routine usually looks like (Wakeup time, naptime & bedtime) as it may be possible to tweak it to both improve her nap and bedtime.
While she is napping for just 1hr i would also bring bedtime much earlier & have her down for nearer to 7pm, it may be 7.45-8pm is still too late (unless she is napping ridiculously late in the afternoon). The more wired she is the harder she will fond it to settle, so going earlier to bed can often mean they settle sooner & sleep longer.
Also does she have a regular bedtime routine? Regular sequence of events to prepare her for sleep may help her settle better.
Finally with regard to pick up put down, once they become toddlers thid can be too stimulating & wind them ip more. At the most you want to just do the put down part, so if she stands & cries you lay her back down, the aom being to do most of the soothing in the cot.
Personally i would address routine first & foremost, i think if you can get her a bit better rested the bedtime antics will settle. Post her routine & id be happy to offer some suggestions.