Don’t You Feel Well? - Illness, Allergies & Emergencies

The overall shape of such a sentence is very simple in English: 'Someone did something to their [body part].'
'Burnt' is an alternative spelling of 'burned'; it is also more like the word sounds in everyday speech, like the past participle ('a piece of burnt toast').
We also use the phrase 'to burn one's fingers' in a metaphorical way, meaning that someone has involved themself in a risky situation and had a bad experience. ('He burnt his fingers financially by investing money into a dodgy* company.')
*Dodgy = dishonest, unreliable etc.: this is an informal term, suggesting (in this case) that if anyone had looked more closely, they would have found cause to be suspicious, and they would have 'thought better of' putting their money into the firm.

'Sick and tired' is the standard phrase. It suggests you are ill (not feeling well), besides being exhausted with the sustained effort of coping.

'Itching' means that there is small-scale pain, which may be very inconvenient but does not really feel as bad as a deep, throbbing ache. It is usually superficial, i.e. the problem is on the outside of the skin, such as when you have been stung by nettles during a country walk.
Inflammation (Answer 2) means that the flesh is warm to the touch (like a 'temperature'), which suggests some quite serious and deeper damage to the tissues of your body, and/or maybe some infection.
'Bruising' (Answer 3) is when there is bleeding beneath the surface, and that part of you may have turned an abnormal colour ('black and blue' is one alliterative traditional phrase).
A body part is 'swollen' (Answer 4) when it comes up in a bigger size, quite possibly from being bruised and/or inflamed. If you have an uncomfortable or unfamiliar 'lump' or swelling, you should probably consult a doctor.

You may experience 'stomach-ache' (it looks a bit odd on paper or on-screen; practice saying it*!), but not the others.
'Heartache' is not a physical pain (though of course, you might have a pain in your chest, and if you do, that could be serious): we use it more metaphorically to express how you feel in a difficult situation, e.g. 'After much heartache, he decided to emigrate'.
Pain in the eyes (e.g. from too much reading, &/or in strong sunlight ~ not so common in Britain!) is more usually known as 'eye strain', while pain in the foot/feet (e.g. from too much walking while sight-seeing) is probably 'sore feet'.
There was an old joke about the (imaginary) British custom of baking a special cake called 'stummer cake', to make someone feel ill in their 'insides': so they would 'have stomach-ache from eating too much stummer cake'.

Unlike in many languages, we do not say 'I have hurt the foot on/to myself', or any such thing.

Answer 2 is also fine; Nos. 3 and 4 would probably be understood (especially in a 'live dialogue' situation, where you could point to them etc.), but are not fully correct English.

They're all OK apart from No.3. We say 'toothache' (with 'tooth' in the singular form), because even if a whole section of your mouth is painful, there is probably only an actual problem with one single tooth.
Other languages might express these ideas more along the lines of 'I have a pain in/at/on my [body-part]', or 'My [pody-part] hurts'. Of course you can say this, because it will be (and needs to be!) understood; but the ones where ~ in English ~ we 'have an ache' are a very useful group of phrases to know.

Three parallel examples here, 'for the price of one', but the overall shape is still the same.
Obviously, clothing is slightly different from actual parts of the body, but we use the same language structure ('he ripped his shirt', etc.).
'Bruising', and the noun 'a' bruise', refers to where part of the body has been hit and changes colour, probably due to internal bleeding (it goes red, purple, blue ... ). We also use this word metaphorically if we talk about 'a bruising experience' ~ this may just have been difficult psychologically, without actual injury. ('He felt he had been bruised by being sent away to boarding school; she had a fairly bruising meeting with her manager about the failures on last month's project.')

We describe this as 'a sprain'.
You might take Aspirin to help to dull the pain: Aspirin and 'a sprain' are almost anagrams.
Otherwise, the problem should be fairly obvious to anyone that can see you; otherwise, say 'My ankle is hurting' and they will probably realise what you mean.
One situation where a visitor from abroad might risk spraining an ankle, is when they are stepping off the pavement to cross a road, and suddenly find traffic approaching quickly from the 'wrong side', so they try to get back up onto the edge of the pavement and trip over. Obviously, we hope this never happens to you!

A 'fiddle' is the informal name for a violin (particularly in folk music, rather than more classical types). The word 'fiddle' also happens to be a verb, meaning 'to play around with something, with one's hands, as a distraction' ~ similar to 'fidget', when (typically) a bored child fidgets ( = wriggles around ) in its chair and maybe also fiddles with something, like a small toy. From this it also means 'to interfere' unhelpfully, and even to be dishonest ('The accountant lost his job once the managers saw that he had been fiddling the books').
Being 'under the weather' (Answer 1: how very British!) or 'off-colour' (Answer 2) are both very common expressions. Obviously one would be careful using Answer 2 if talking to 'a person of colour' (i.e. someone of African, Indian or perhaps Asian heritage); but certainly, for a 'traditional Brit' with pink Caucasian skin, they may indeed look a slightly different colour if they are not well: greyer, whiter, pinker (or whatever) in the skin, and enough so to be noticeable.