Battlesuits already exist in Traveller, and are what is called "Battledress". The Central Supply Catalog has some varieties of Battledress that expand the basic options somewhat.
Mechs can probably be built with the new Vehicles supplement, but I don't have that supplement, so I'm only guessing.
This table is, indeed, calculated by assuming that a ship can produce constant acceleration away from its origin, instantaneously pivot \$180^\circ\$ at the midpoint of its journey, and constantly decelerate for the second half of its journey. The greatest veloticy thus attained (w.r.t. origin) in the table above is roughly 1 million m/s--approximately 1/3 of one percent of c--so we're comfortable sticking with classical calculations.
The formula that produces the above times is:
$$ t = 2 \times \sqrt{\dfrac d a} $$
where \$t\$ is measured in seconds, \$d\$ in meters, and \$a\$ in meters/seconds2. (G is rounded to 10 m/s2 for ease of calculation. Note that the table gives distances in km, so you've got to tack on three zeroes to end up in meters.)
Those who would like a refresher on their classical kinematics, read on:
Recall that the distance travelled when starting at rest and undertaking constant acceleration is given by
$$ d= \frac 1 2 at^2 $$
Solving for t gives us
$$ t=\sqrt{2 \times \frac d a} $$
In our situation we consider the time (t1/2) to accelerate to the midpoint of the journey (d1/2):
$$ t_{\frac 1 2}=\sqrt{\dfrac{2 \times d_{\frac 1 2}}{a}} = \sqrt{\dfrac d a} $$
Doubling this gives a total trip time of
$$ t=2 \times \sqrt{\frac d a} $$
Best Answer
Mild spoiler alert: The very first scene of The Fall of Tinath (published 2017 and released with the Core Rulebook as part of the Traveller Starter Set) has a situation where player characters need to contain and extinguish a fire on the ship they are on. The main purpose of the scene is to familiarise new players with the task system for Traveller.
Succeeding on a test reduces the Fire Value by 1 (it starts at 3 in this scenario and is extinguished when it reaches 0), while failing a test has a 1-in-3 chance of increasing the fire value by 1.
There may be further rules on firefighting in Special Supplement 4: Rescue Ops (published 2014). I do not own a copy, but looking at the last page of the preview on DriveThruRPG, the Service Skills table for the "Rescuer" career (which includes firefighting) lists the following skills:
A character who has done their first term (basic training) in this career will only have level 0 in each of these skills, which presumably includes how to fight fires. The presence of Battledress on the list (which is no longer a separate skill in the current version of the rules, being subsumed into Vacc Suit) suggests that this career description is only for rescuers on very high tech worlds - see speculation below.
The following are my personal opinions. The Fall of Tinath rules are usable, but the referee should select appropriate eligible skill/s based on the context of the fire. For fires anywhere on a spacecraft or space station, I would definitely include Vacc Suit as a useful skill. (The failure to include Vacc Suit in The Fall of Tinath skill list is especially odd because the purpose of the firefighting task is to give other characters something to do while the pilot and engineer are occupied with other tasks!)
Obviously, the presence or lack of appropriate firefighting equipment and ability to access the fire should increase or decrease the difficulty of the firefighting task. Realistically, there should be a Profession (firefighter) skill that can be used for any firefighting situation, though it would be used so infrequently by player characters that it is understandable that it is not included in career skill lists - even ex-rescuers would consider it to be a "wasted" skill.
Ultimately it is up to the referee to earn their (generally non-existent) pay deciding the rules for any given situation. If the fire is in a TL4- settlement and the only effective way to fight it is to form a bucket brigade then it may be that Leadership and/or Tactics are the key enabling skills to get people organised, with individual heroic efforts using Survival or Athletics (Strength or Endurance). If the fire is in a TL15 metropolis with computer-controlled firefighting equipment then Electronics (computers) may be the most important skill.