No, you do not. About 2/3ds of the way through (and especially after the Zeratul missions which give you a great deal of research poitns) you cap out on both, and you start being able to "sell" extra research to the Moebius Foundation for 10k each.
The equation is very simple for armor (much simpler than Warcraft 3 for example).
Each unit has a number of attacks. Zealots for example hits 2 times for 8 damage each shot
Each unit has a number of armor points. Each armor point reduces damange by 1 by attack
Each race can upgrade armor (protoss shield works the same way). Each armor upgrade gives +1 armor.
So if a zealot would hit a zergling with carapace upgrade 1 (Total armor = 1) the calculation is as follow
(8 - 1) x 2 = 14 total damage
Now Starcraft 2 units gets bonuses versus particular types of units
Suppose that a Viking (2x6 +8 vs armored) would try to shoot down a battlecruiser (3 armor). The calculation goes like this
(6 + 8 - 3) x 2 = 22 damage
So you can say that multiple hits is a weakness, since the armor gets deduced multiple times.
Now for the second part of your question, is carapace upgrade worth it? I'd say yes, but the real answer is "it depends on when to get it". Starcraft is a game of tradeoffs, so getting upgrades is good, but losing because your enemy has more units/firepower than you is bad.
The question becomes "when is it safe to upgrade?". As a rule of thumb, I'd say that whenever the game gets lengthy, that you can secure an expansion for the extra gas this cost, do it.
Best Answer
+1 Upgrades just do exactly what you would think they do. They increase the attack damage by a certain amount. For your marine it is one damage, so it then deals 7. The same goes for armor upgrades. A zergling with +1 armor will only take 6 damage when shot by a marine with +1 attack. Of course there might be other buff/debuff effects for example a Guardian Shield from a Sentry which is basically +2 armor.
While armor upgrades always decrease the damage taken by one, the exact effect of the attack upgrade depends on the unit. For marines each upgrade increases attack damage by one, but Ultralisks for example gain 3 attack damage per upgrade. (Thanks to @OrcJMR for pointing that our in a comment)
Those upgrades have a huge impact on the game. The player with the upgrade lead has a big advantage in fights, assuming that the army supply is around even.
The prime example it the matchup between Zealots and Zerglings. With no upgrades a Zealot needs 3 hits to kill a Zergling. With +1 attack the Zergling dies in 2 hits. If the Zerg player then get +1 armor a Zealot needs 3 hits again. The same logic applies for all other cases as well even though it might not be as impactful as in this example (but it often is).
Good players always try to attack the enemy when they have an upgrade lead. This is often refered to as hitting a timing. Hitting a timing when your +2 is finished but the opponent's is not can decide the fight in your favour.