I have heard/read there is a lot of confusion regarding who was martyred in the dungeon after the martyrdom of Imam Hussain (as).
Was it the young girl of 4-5 years of age janabe Sukayna or janabe Ruqayyah? Also there are narrations that Imam Hussain (as) had 3 daughters. While some have said He (as) had 2 daughters. That is janabe Sukayna and Ruqayya is the same daughter of Imam Hussain (as).
Would be grateful to clear the confusion.

Sakina was the teenage daughter of Imam Hussain (as). Ruqayya is the 3-4 year old daughter of his. She is the one who died in the dungeon of Yazid. Sakina is also buried in Syria as well. The urdu speaking community refer to both of these daughters with the same name which is why there is confusion. There was another daughter who was left in Madina because she was ill.

Is playing UNO allowed. It is very similar to playing cards?

agar society ise cards me hi shumar karti hai to haram hai chahe baghair shartbandi ke ho, agar aam public ise taash me nahi jodti to bina shart bandi khel sakte hain

Is it haram to play with Pokemon Cards or Uno Cards since cards resemble playing poker?

Wa alaykum salam. No this is not haram

Fajr time was at 3.44am today. I was eating until 3.43am and when there was around 10secs left I intentionally swallowed the saliva at the back of my mouth which contained melted chocolate full well knowing that I may probably run out of time whilst swallowing this.
The first swallow was within time yet the second swallow that followed (didn’t really have control over it) happened when the clock hit 3.44am.
Is my fast valid?

You should give yourself time before fajr which is called imsak where you stop eating and drink prior to fajr by 10-15 minutes. In your case now you should consider your fast valid but as ihtiyat pay kafarra of feeding 60 people. This can be done anywhere as an example this would cost $30 in Afghanistan or less in Iran. Also as ihtiyat to do the qadha

How can one increase their hope and imaan in Allah taala and think good of him?

Read the stories of the prophets of Allah (as) and these would instill hope and faith inshallah. I would recommend

https://imam-us.org/wp-content/uploads/2020/12/Gods-Emissaries-Published-by-IMAM.pdf

Can we get corona virus test while fasting?

Yes

Can a believer also loose his tawakkul sometimes and become negative?

Faith is something that can fluctuate in people but we always need to work on strengthening it by reading Quran, narrations of Ahl ul Bayt (as) stories of the prophets (as) and Ahl ul Bayt (as) for example among other things

Is it permissible to use mouthwash in Ramadan?

Yes but do not swallow any

Recently, I was pondering upon my relationship status with my IMAM atfs
How will I know how close I am to him or he is to be? How could I check if I am pleasing him or breaking his heart? I don't know. What should I do to become a better individual and serve him?

Strive to be the best follower of him by learning about him and implementing his teachings and those of his forefathers (as). With knowledge comes certainty.

Do we know where animals go after they die? Obviously they go to Allah SWT but do we know what specifically happens to them? Can we see our pet animals in Jannah?

Surah al-an'am, ayah 38 and Surah al takwir ayah 5 show that the animals will also be gathered.

I've heard it a lot from Sunnis and I'm sure that we have the same belief on this but I wanna confirm. Sunnis say that RasoolAllah Prophet Muhammad (SAW) said it's better for us to get stabbed with iron needles than to touch a non-mahram. Is this hadith true? Also if a non-mahram goes to try to hug us or try to shake hands with us then is it permissible if it's our aunty or cousin? If not then how can we prevent such engagement.

I have not seen this particular narration but inshallah these following should give some clarification

Imam Muhammad al-Baqir (as) said “Those who look at a non-mahram woman with an evil intention and fill their eyes with this sight (that is they look for a long time) Allah will, on the day of Qiyāma fill (pierce) their eyes with burning rods. They will continue in this way till Allah completes the hearing of the cases of all the people. Only after that will He order them to be thrown into Hell.” (Greater Sins by Ayatollah Dastaghayb Shirazi)

Imam Muhammad al-Baqir (as) said “There is no man who is not to some extent guilty of fornication. The fornication of the eyes is the evil glance, the kiss is the fornication of lips and touching a non-mahram woman is the fornication of hands.” (Greater Sins by Ayatollah Dastaghayb Shirazi)

The Holy Prophet (s) said; “The wrath of Allah is very severe upon the married woman who apart from her husband sees another non-mahram man with a prolonged glance. When this woman does this, Allah the mighty invalidates all her good deeds and does not give her any rewards.” (Greater Sins by Ayatollah Dastaghayb Shirazi)