Brother Alaeddin,
Salamun Alaikum.
I see your question in two parts so I will answer accordingly.
ARE TATOOS HARAM?
Haram is a very strong word and one must be careful not to pronounce something 'haram' if it is not explicitly mentioned or strongly implied by the Quranic verses by strong analytical deduction.
Albeit the Quran does not give direct references to 'tattoos', it does give much guidance from which we can derive an answer.
I personally would not consider 'tattoos' or any decorative items on the human skin inherently taboo.
It is the 'nature' of the tattoo or 'decorative impression' (and possibly where it appears on the body) which is of more relevance. Tattoos can be graphically lewd, gross, profane, blasphemous etc which would run contrary to many of the Quranic verses and would amount to a transgression. This would clearly be taboo. Permanent tattoos can also mutilate the skin and harm it in a manner which would arguably put it at tension with verses 4:118-119
004:118-119 (Part)
"...and he said "Most certainly I will take of your servants an appointed portion: I will surely mislead them, and I will create in them false desires; I will order them to slit the ears of cattle and to deface the creation of God..."
I would find such a permanent affliction on the human body difficult to support and would personally strongly discourage it in light of the Quran.
However temporary tattoos much like Henna for women which are not permanent on the skin should not be a problem as long as they do not amount to a transgression. However, decorating the skin in sensitive areas, or private areas of the human body is arguably contrary to the Quranic teachings as more than likely a 3rd person would be involved to impression it.
As far as images and statues being haram, I find no support for this in the Quran. In fact, I find such a concept negated. Prophet Solomon had 'statues' made for the Lord's temple (34:13). The difference is he did not make them to worship them but to decorate the temple. (2 Chronicles 3:1, 3:10). In contrast, Prophet Abraham's father worshipped the 'statues' (021.051-052). So it is not the 'statue' which is inherently 'haram' but what you do with it. There is a difference. However, lewd graphical images or statues which run counter to the spirit of the Quran's guidance are obviously a transgression.
DEATH FOR HOMOSEXUALITY
Punishment of death is only actionable in two circumstances in the Quran.
(1) As a retribution for causing 'fasaad' (gross mischief / evil, beyond all bounds) in the land (punishable by the state)
(2) As a retribution for murder (punishable by the state).
There is no punishment for death for homosexuality in the Quran. However, the Quran deems homosexuality as a sinful transgression, a 'fahisha' and it attracts condemnation and some form of retribution if one does not repent (for an Islamic state to decide based on circumstances). Verse 4:16-17 makes this clear.
004.016
'And as for the two of you who are guilty thereof, then punish them both. But if they repent and amend, then let them be. Indeed! God is Oft-Forgiving, Most Merciful.
Homosexuality in the Quran runs contrary to the nature of humans where genders of opposite sex have been created for intimate relationships and procreation through 'lawful marriage' (Nikaah) . The practice of homosexuality has also been severely rebuked in the example captured in the narratives regarding the People of Lot.
007:080-81
"And Lot when he said to his people "Do you commit lewdness (Arabic: fahishata) such as no people in creation (ever) committed before you? Indeed, you approach men lustfully in preference to women. Nay, you are a people transgressing beyond bounds / committing excesses"
I hope this helps,
Joseph.