@Mantidkid Sadly most crickets straight from the pet store are malnourished, and hungry to the point they are highly aggressive and turn on each other and possibly mantids or other pets too. Indeed already dead or dying crickets make easier prey for the living, so it can lead to bacteria and other nastiness.
Thankfully not all pet stores are so bad, but most tend not to spend money feeding crickets until they are sold. With few stores only giving them water or water gel crystals when the crickets are sold. To combat the issue, the crickets should be well feed and watered at least 24-48 hours before using them as feeders. Not only does it increase their potential use as a feeder with good nutrients, it makes the crickets much less aggressive.
If though if the mantid has been without feeders, or the keeper does not know of the potential danger, a few cricket feeders usually does not affect a healthy mantid.
Depending on the amount of crickets you may need several feeding and watering containers for them all to have access. In the photos above it appears you do not have enough space for the crickets to access food/water. I personally use two 5" containers for water, and another for food, but then again I use 56 quart or larger storage containers for cricket tanks.
To remove dead crickets using tongs/tweezers works great and keeps your fingers clean.
Although I imagine you are asking how to get the crickets from the container to the mantid habitat easier.
Regarding from the pet store to your cricket container, all pets stores I've bought from, put crickets in plastic bags that easily shake into my containers (or a funnel can be used too if needed). If you get crickets in boxes or other materials, you best option is to put them into the cricket tank inside a large container to prevent escapes - a decent sized bathtub works in most cases. Now how to easily get crickets out of their container to use as feeders depends on a few things.
For example if you bought the Kricket Keeper type containers to house the crickets, they have removable tubes that the crickets hide in. To use those just pull out the tube, blocking the bottom of the tube and pour out a few into your mantid habitat. For other cricket containers you can simply use tongs/tweezers to capture crickets, being careful not to crush and killing the crickets. After some experience it can be a rather easy task, I do that myself with my 10" tongs when I have only a few hungry mouths.
A much easier way is to use a small holding container, I find the empty Walmart cotton candy tubs work great. Those tubs are about 4" round base and about a 12" tall preventing crickets from escaping. With the holding container simply shake the cricket egg crates, cardboard tubes, etc into the container while held over the cricket tank. Then you can now easily grab the crickets from the holding container with tongs/tweezers, fingers (not recommended as you can easily crush the crickets and they can bite), or shake a few crickets into
a funnel in your mantids feeding hole.
If you have any questions just ask, I've used them for years and mine successfully breed in large numbers keeping their colony/culture self-sustaining. The subject is one that I have started for another guide, but will be awhile until it is posted.