I am going to raise my query in the form of points. So here we start!
1. Suppose I am having a story and a task allotted to my name. Now, if I am working on the task then should that story be in progress?
2. Same condition as referred to in the above question. Now if there are 2 tasks 1 alotted to me and another to the developer, if I completed the story early(marked done) and the developer completes in the 3 next days then will that story be in progress for that much log time. If yes then can I start with another story without fearing about that a single user cannot have 2 stories in progress at a time
3. Is there any way by which I can check the amount of time invested in a particular story?
When you say tasks, do you mean sub-tasks? If so, as long as some work for the corresponding story is in progress I'd move it to the 'in progress' status. If you expect that one or more of the (sub)tasks will be in progress for (much) longer than yours, maybe you can create separate stories and link them using dependencies. In my experience, the process works better when a story can be completed within a sprint, and we try to size it that way.
Regarding point 3), this other post may help: https://community.atlassian.com/t5/Jira-Core-questions/How-can-I-display-inside-each-story-Total-Time-spent-by-Each/qaq-p/258273
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