I agree with what everyone said above - I have just seen their posts, so there will be some repetition below.

Allchopin is right: I mean 1/60 minute (= 1 second), not 1/60 second (I have corrected it in the original post.
1. Giving Crochet 1 count we get Semi breve = 4 counts, Minim = 2 counts, Quaver = 1\2 count so what is a 3 count one called?
There is no three count note value, because every note value doubles or halves the others. So if crochet = 1, minim = 2, semibreve = 4, quaver = ½, semiquaver = 1/8 and so on.
So, how can one write down a note value that equals 3 counts? There are a number of ways. The most used is to “dot” a note. The dot adds half the value of the note. So the minim = 2, a dotted minim will equal 3: dotted minim = value of the minim (2) + half the value of the minim (1). A final way is to use triplets, where for instance, a quaver triplet (consisting of three notes) equals the time of two normal quavers (consisting of two notes), as allchopin mentioned.
A dotted semibreve will value 6: dotted semibreve = value of the semibreve (4) + half the value of the semibreve (2)= 6.
Now, it is important to remember that all these values are relative. I am assuming that crochet = 1. But you can give to the crochet whatever number of beats you want, but then all the other note values must change accordingly.
But if crochet = 1, then a dotted minim = 3. Another way to get 3 beats is to “tie” a minim and a crochet. (or to tie three crochets).
A final way is to use triplets, where for instance, a quaver triplet (consisting of three notes) equals the time of two normal quavers (consisting of two notes), as allchopin mentioned.
There are rules for when to tie and when to dot, and they will depend on the time signature. If you want the details, I suggest you get a copy of
Eric Taylor – The AB guide to music theory (ABRSM).
2. If crochet is assigned 60 then for that whose sign is A DOT AFTER MINIM i.e the third count value note will get what value? I read two different ones, now how can that be possible that crochet being 60 yet the note gets the value of 54 one time and 50 second time?
As I said above, a dot adds half the value of the note to the time value of that note. So if you have a dotted minim with a tempo of crochet = 60, all it means is that you must hold the dotted minim for three beats of the metronome.
I am not sure what you mean by the note getting the value of 54 and 50. (unfortunately I do not have John Thompson’s book,but I will have a look at it on the library this weekend).
How can I insert Staff Symbols here in Forum correspondence.
I don’t think you can. I think the only way is to scan (or create a jpg/gif file of it using a notation program) and host it on a website like tinypicture, and then insert the URL in between the image command: [img ]https://URL[/img ].
Have a look here where this is explained in more detail:
https://pianoforum.net/smf/index.php/topic,5503.0.htmlBest wishes,
Bernhard.