The tip leaves are touching the potting mix. Will this cause rot? Should I remove the tips? I had removed the tips the last time I grew these grocery store cuttings. The potting mix is fresh and soilless.
Those miserable looking things are unlikely to grow.
Try taking cuttings at least 3 inches (8 cm) long, and keep them in a pot of water until they start to grow roots.
You can remove some of the lower leaves if you like, but never remove the tip of the stem from a cutting. Without going too far into plant biochemistry, the growing point at the tip of the stem produces hormones which increase root growth, and the roots produce hormones which increase tip growth.
That is how a plant keeps a balance the amount of roots and leaves it grows. If you cut off the tip of a cutting, you are telling the plant not to grow roots!
I would always start mint in water - any available container that’s vaguely the right size should do. Fresh mint is very eager to root - I once forgot a water bottle with a sprig of mint on my desk at the office and by Monday I saw the first rootlets.
It is important that you use the freshest mint possible - the longer it was laying around in a store the smaller the chances of success, especially for those herb bundles that are sold as refrigerated cuttings in a plastic baggie in the vegetable section. Pick the best, crispest you can get, trim the ends and lower leaves (to minimize evaporation) and dump the whole bundle in water like you would for a bouquet of flowers. If you can get potted mint, choose those. Place the container out of direct sunlight in a light spot. I use my north-facing kitchen window, because the light is good and I see the cuttings multiple times a day. If over the next days you see sprigs wilting, remove them immediately, keep the good ones. With a bit of luck, you’ll have roots in a week or two.
If you want to root the cuttings directly in soil, remove the lower leaves (as you did already) and poke holes into the soil so that the majority of the stem fits in, leaving just the top third or so with the leaves. You want to have a good portion of the stem in the soil, so that you have plenty of places where new roots can grow and both anchor and feed the plants quickly. A bit of cling film or other transparent cover can help reduce evaporation in the first few days. If your cuttings came already a bit wilted, place them in water for a few hours before planting to give them a chance to rehydrate. (But then you could just as well leave them there and proceed as in my first suggestion.)