Is there any way to find a specific field research? For example I am looking for hatch 5 eggs & cannot find it in my area

Facebook/WhatsApp/Discord groups. In my area both the Facebook chats and discord have quest reports where people post research tasks they find. Other than you’ll have to just keep searching

Some public gym/stop maps have also created a functionality where users can mark the quests on them.

Hatch 5 is very rare, my neighborhood has about 50 stops and that quest has been on one of them exactly 3 times since quests began. Sometimes a day goes by without anyone finding one in all the thousands of stops in my city.

Worth of mouth. Get in touch with your community. If someone finds these special quests, they will share in the local groups. That's the only way.