All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
We shall answer your questions in the following brief points:
Firstly, if the image of a face is without a body, it is not included under the prohibited images. Ibn Qudaamah wrote, "The same applies to incomplete images, such as the image of a body without a head, or a head without a body, or an animal's head attached to the body of a non-animal; this is not included in the prohibited images because it is not the image of an animal." [Al-Mughni]
Secondly, there is no harm with the images of animals that are defaced in such manner that their features are unclear. This applies to the bird in the Twitter logo. Ibn Nujaym said, "...likewise, if the image is defaced, it is the same as the image without a head." [Al-Bahr Ar-Raa'iq]
Thirdly, the authentic Sunnah prescribed the removal of the cross or the like of prohibited symbols. ‘Aa’ishah reported that the Prophet, sallallaahu ‘alayhi wa sallam, never left anything in his house containing the image of a cross without obliterating it. [Al-Bukhari]
It should be noted that scholars held different opinions regarding the ruling on keeping and using items bearing such symbols; some held that it is disliked, while others held that it is prohibited. The author of Kash-shaaf Al-Qinaaʻ said, "It is disliked to have the image of a cross in clothes and the like, such as hats, coins, rings, etc. This is because ‘Aa’ishah said, 'The Messenger of Allah, sallallaahu ‘alayhi wa sallam, never left anything in his house containing an image of a cross without obliterating it.' [Abu Daawood] The author of Al-Insaaf said, 'It is possibly prohibited, and this is the stronger opinion.'"
If a Muslim cannot remove this cross and there is a necessity or need to use the item bearing its image, then we hope that there would be no harm in using it, following the view of the scholars who held that it is disliked (not prohibited). Some scholars held that it is allowed to use the legal concession in the case of need in order to ward off hardship.
Fourthly, assuming that the owners of these mentioned companies promote unlawful matters, this does not entail prohibiting the use of their lawful software. The authentic Sunnah mentioned dealing with the Jews although they might use their money in unlawful ways and in acts of hostility towards Muslims. ‘Aa’ishah said, "The Prophet, sallallaahu ‘alayhi wa sallam, died while his armor was pawned to a Jew for thirty Saaʻs (volumetric unit of measurement) of barley." [Al-Bukhari]
Nevertheless, if the Muslim community decided in some cases that boycotting some of these companies serves as a deterrent to them against their unlawful practices, then the Muslims should implement their decision. Boycotting may be obligatory if it proved effective in deterring them from their evil acts.
Allah knows best.