PQ are a gc.com thing and not a pgc. The only pq functionality on pgc the PQ Splitter that gives you interval to create PQs to get all caches from selected regions/countries. If you would create intervals that result in more then 1000 caches the PQ will still only contain 1000 caches and you don't get all caches in the area.
The date filter for PQ are based on hidden date and not publish date. A small but sometimes important difference.
If you are a paying member on pgc you can add caches to a vgps and create a gpx from that list. You can get 10000 caches per day that way. That is almost the same thing as a PQ.
Using the the hidden date intervals and use the dates for 10 PQ and adding that to the vgps is an option.
The number of caches is the same at you can get from gc.com in a day with PQ:s
You can always do both and get 20000 caches per day
Unfortunately there is no way to get all caches published after a specific date in a PQ or on pgc. All caches with hidden date after your last query is possible but that is not the same. Some overlap is needed.
The last paragraph in not strictly true. The gc.com API provided a function to get caches published in a interval of max 30 days. GDAK can do that on but only in 50 km circles and the last 30 days. GCdroid uses in in FTF search.
The only way to do that in a whole county/regions that I am aware of is to use GSAK. And if you what to have a comprehensive offline list of caches using GSAK is probably the best option. Then you also can use the API to get another 6000 cache per day but the filter alternatives are a bit different there so you have to be a bit creative to get all the data as efficient ass possible