In any case, every person's circumstances are different.
1) Taking an interview does not obligate you to take the job. Taking an interview helps
you learn your status. Receiving a job offer helps you determine your market value.
As long as you keep in mind that starting a new job does not mean you will be able to
keep that new job.

2) You are a professional, hopefully, and if you accept a counter offer you need to do it
in a professional manner. This means an actual contract in exchange for giving up an
opportunity elsewhere.

3) Taking a "sick day" to interview is not all that professional. Taking a personal day is.
What you do on your days off are your business.

There are ways of leaving that do not burn bridges, assuming you are working with
professionals. If you use the offer, that you have not yet accepted, as leverage, that
may be risky but it's what professionals do. Don't make it personal.

An unwillingness to look at options and negotiate generally just guarantee you lower
pay over the long haul. If nothing else, use the offer to give you a little backbone to
address the issues that are making you unhappy. If you can't resolve those issues
then it is time to move on.

Once you accept an offer, that's it unless you want to burn the bridge with the new
company. Likewise, letting your current employer know you are being recruited is
not a bad way to accelerate the process of getting better compensation.

"I'd like to stay, but the new place offers healthcare/daycare/matching 401K, etc."
If you have multiple offers its even better.

Just remember, in most places you are not "family" you are simply an employee.
If they are not treating you as family they have no right to demand such a thing
as "loyalty" especially if they are willing to cut your benefits and yet give the top
brass bonuses.