No, you don't have to be a Mason to be Mormon; few Mormons are also Masons.
I only know of two temples that have any supposedly "Masonic" symbols. The "Masonic" symbols actually predate the Masons by several thousand years. In fact, they can be seen on the Egyptian papyrus that Joseph Smith bought from an antiquities dealer. The text was a book of breathings, a type of Egyptian endowment, similar to the Mormon Temple endowment.
No you don't. I have been a Mormon, but I have never been a Mason. In fact, my entire family, and pretty much my whole community, is Mormon but I don't know any Masons.
However, many of the early leaders of the church were also joined at one time or another with the Masons, including Joseph Smith, the founder of the Mormon religion. Masonry is further linked to Mormonism because much of the Mormon temple ritual is similar, if not identical, to Masonic rituals. These are more than just similarities between symbols on some Mormon temples.
Some Mormons explain this by saying that the Masonic tradition dates back to the time of Solomon's temple. They say that God's truths were known at one time, but perverted. They believe that Joseph Smith restored these traditions to their full correctness. (I don't have a source for this, except my personal family and Sunday school teachers. This is what I was told when I questioned it myself after reading information on line.)
Other Mormons don't know about the similarities between their rituals and those of the Masons. This is because it's not widely talked about in the church and it's not a common instinct to say to ones self, "Hmmmm, I wonder what my religious ceremonies have in common with Masonic ceremonies."
It is interesting to note that the Masons themselves date the origin of their tradition one of two ways. The first is that Freemasonry began in 1717 in London. However, some say that this is simply when the first lodges were divided and that the tradition began earlier. However, the earlier date is still much later than the time of Solomon. According to Masons, the date is sometime between the death of Robert Cochrane in 1482 and the death of the Stuart King James I in 1625.
No you don't have to be a Mason to be a Mormon. And believe it or not those are not Masonic symbols, or rather they are masonic symbols but they are not special only to masons. The shared symbols are coincidental.
Masons study symbols and they have a lot of them. Their symbols can be found in many religions. The Arabic number system, the one you learned in elementary school, those numbers are masonic symbols. The jolly roger (pirate flag) is a masonic symbol. The crucifix found on most churches is a masonic symbol. Each symbol has meaning just like each letter of the alphabet has meaning. The meaning of some symbols are less popularly understood since our society has become more literate.
Masons aren't about one particular faith, so while there are Mormon Masons, Mormons aren't any more likely to be Masons than people of other faiths. It certainly isn't required. Joseph Smith was a Mason for a short time, yes, but he left the organization shortly after joining, citing an amicable difference in values. And while there are similarities between Masonic and temple symbols, they're not from the Masons. The dates for the building of the first temples and Joseph Smith's membership don't really match. There are books written on the subject if you're that interested, though even I'm not.
Still, even if those symbols were exact copies of the Masons', what would that really mean? The Masons have taken symbols from many cultures and times. Of course they're bound to overlap at some point with current religions, especially Christian faiths (as the Mormons are), because even if the term didn't pop up until after Christ's death, there have been Christians for thousands of years. It doesn't mean anything.
Joseph Smith took some stuff from Freemasonry, like handshakes and tokens he claims you need to get into heaven. he even has the apron and special cloths. The all seeing eye, the handshake which is also on the outside of most UT temples, however UT doe shave a masonic temple.