To put it simply, the difference is:-
If you want to say -- "Have they arrived already?", you use "他和她一起来了吗?" -- meaning you are just inquiring as to whether they have arrived, nothing more. The "了" here indicates "already?", i.e. "completion"
But if you want to say -- "Did they arrive together?", ...
Let's make it to a affirmative sentence for analysis. 他和她一起来的 vs 他和她一起来了.
的 here is a modal particle denoting affirmation. 他和她一起来的 means he and she together came here. You can insert a 是 as indicated by other answers but do not have to because 是 might add more emphasis than without it if you don't speak in a casual way. 他和她是一起来的.
了 in 他和她一起来了 denotes a ...
A very special function of 的 is to emphasize, especially together with 是。You can notice that many political big shots like to use it: 中国是绝对不会主动侵犯人家的。 Using 了 is simple story telling. Using 的 is emphasizing: it's "HE", and it's "WITH HER", not anybody else. BTW, I sincerely wish you understand because I didn't use any grammar jargon at ...
Here is a similar question and answers: 了 vs 是…的 as “factuality markers”
他和她一起来了 = he and she have come together
他和她 is the subject, 来 is the verb; the verb particle 了 indicates the completion of the verb 来
他和她 [是]一起来[的] = he and she, [are the two who] come together
"是一起来的" is the fact that describes the subject 他和她
This doesn't solve the root issue of not being familiar with enough characters, but if you often have trouble reading small Chinese text online, you can install a Chrome/Firefox extension called Hanzisize which will allow you to enlarge Chinese Characters without changing other elements on the page.
I'd probably put:
加 is fine too if what you cook is a type of 饭, not 菜.
In Chinese, if the main ingredient is rice, it's called xxx饭 such as 卤肉饭，菠萝饭，etc. Otherwise, it's called 菜.
If you want to express that you mixed up every thing including steamed rice, you can say:
做"name of your dish" 时(or 我这种饭的做法是)，我会把番茄，西兰花，...
Clarification by the OP
I place all the vegetables in a pot with rice and cook them. I think 加 would indicate I add them in after cooking the rice
In this case, 煮饭 means 'cook rice', 加 would mean you add them into the raw rice when you cook them
The sentence would be
我煮饭時会把番茄，西兰花，胡萝卜和辣椒加在米中一起煮 (when I cook rice, I would add tomato, broccoli, carrot, ...