You're doing a one-tailed test with the coin, even though the original question of interest was about fairness, for which the alternative would not be about only a preponderance of tails.
That is, you seem to have chosen a specific direction to test based on what you found in the data -- the same data you want to use in the test. This means the correct p-values will not be what you calculate. Such data-snooping is a particular form of p-hacking. See https://en.wikipedia.org/wiki/Testing_hypotheses_suggested_by_the_data for specific discussion of this issue.
Your associate is correct - if your wanted to test fairness of the coin, you'd double that p-value calculation you did.
The easy way to see the issue is to simulate running a large number of experiments when $H_0$ is true and calculate that your rejection rate (at some chosen significance level) is considerably higher than the one you specified when you choose the direction of the test post hoc -- though in the coin example you could do the calculations by hand easily enough. You can avoid specifying a significance level and do it directly with the p-value if you keep in mind its definition.
Similar comments apply to your dice example (you're choosing which specific face to test for an excess of by looking at the data); formulated in terms of just that test statistic, yes you would multiply by 6.
However there are many ways for dice to be unfair that your specific example glosses over; it's only going to tend to have good power if the specific form of unfairness is a higher proportion on a single face. If for example, you might have wanted to conclude that the die would be unfair if your die was biased so that it likely to produce an excess of all the values between 4 and 6, or if it was biased to produce very few 1's, then your test statistic may not be a good choice; it won't have very good power against those kinds of biases. Your specific test statistic would make sense of your original question of interest had not been about unfairness in general but about that very specific form of bias, before seeing the data.
A common omnibus test for fairness against all alternatives would be the chi-squared test (multinomial goodness of fit).
Typically, also, you'd want a lot more than 10 tosses to test a die for fairness unless you're only concerned about huge effect sizes. (If your sample sizes will be very small, then it does help to pre-specify a specific alternative if that makes sense to do in the context.)
I'd suggest something well over a hundred tosses, and likely considerably higher - perhaps well into the hundreds. Similarly with the coin; ten tosses will only pick up very severely biased coins/coin tossing methods. I'd look at perhaps something closer to a hundred tosses; even then you'll tend to miss deviations that you might find of concern, depending on why the coin needs to be fair.
The correct procedure is to formulate your null and alternative hypothesis before you even collect the data you want to use for inference, so that the data cannot influence what things you decide to test. If you really want to use some data to choose your hypotheses, you then use different data (data which did not participate in the choice of hypothesis) to test the hypothesis.
