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Partial answer:

UPD. by @yalov

1. • Do some (especially borrowed) words in the Belarusian language use the "Russian Г" sound (Гузік)?
   • It is considered outdated to use the letter Ґ (Ґузік) for that sound, isn't it?*

   Consensus on this has never been reached, There are 3 answers:

• Jan Stankievič promoted both the reintroduction of the practice of pronouncing Latin [g] in newly assimilated words, and the adoption of the letter Ґ to represent it.
  Станкевіч Я. Гук "Ґ" у беларускай мове. Бацькаўшчына, 1956

  Also, original norm of Taraškievica (1918) doesn't include Ґ.
  Ґ were discussed at Belarusian Academical Conference (1926), where some changes of the Belarusian alphabet were proposed, but never were implemented; and Ґ is included in modern normalization of Taraškievica (2005).

• Зычны гук Г выбухны has the list of words with [g], but uses Г-letter for that.

• [g] tends to move into [ɣ], and it was done completely for the last 100 years. So there isn't [g] and Ґ.

Partial answer:

2. • Do Ukrainian Г and Belarusian Г have the same pronunciation?

     In general — no.

     Belarusian Г is usually considered to be voiced velar fricative [ɣ].

     Exact nature of Ukrainian Г is debated. Some claim it is voiced glottal fricative [ɦ], while others — voiced pharyngeal fricative [ʕ] (both sample sounds in Wikipedia represent Ukrainian Г badly). Also pronunciation of Ukrainian Г may vary, depending on word etymology and speaker's dialect, sometimes it may turn into Belarusian-like [ɣ]. But most agree that in the "canonical" dialects of Ukrainian language in majority of words (excluding cases of borrowed words when it's sometimes pronounced as [ɣ] just because of mimicking pronunciation of original language, e.g. Greek) it differs from Belarusian Г.

     ^{(However in mid-schools Ukrainian children are usually taught that Ukrainian Г is a voiced counterpart of the Х [x]. Not because it really is (it is not; Belarusian Г [ɣ] is), but rather because no other Ukrainian consonant fits this role better — and such a simplified view is enough for mid-schools.)}

     ^{(Please don't be confused with the "[ɦ] vs. [ʕ]" dilemma. The "[ɦ] vs. [ʕ]" is more theoretical question about right classification of the sound, not about right pronunciation. Per my individual perception, Wikipedia samples for both [ɦ] and [ɣ] represent Ukrainian Г inexactly, and for [ʕ] — even worse (thus Ukrainian Г for me sounds like something between Wikipedia samples for [ɦ] and [ɣ]). Real pronunciation of Ukrainian Г can be heard, for example, in the O. Avramenko's lessons — he begins almost every his lesson with "доброГо дня".)}

   • Do Ukrainian Ґ [Belarusian Ґ] and Russian Г have the same pronunciation?

     In general — yes. It is voiced velar stop [ɡ].

   More info: Ukrainian phonology.

3. There is "alternative sound of Г" in some words of the Russian language (ага, угу, бог, господь). What type of Г is it?

   The "Russian phonology" Wikipedia article says it's either [ɡ] (strict), or [ɣ] (like Belarusian).

   While the "Ge (Cyrillic)" Wikipedia article says it's region-dependent: /ɡ/ generally, /ɣ/ in the Southern Russian dialect and may be /ɦ/ in the regions bordering Belarus and Ukraine.

Partial answer:

2. • Do Ukrainian Г and Belarusian Г have the same pronunciation?

     In general — no.

     Belarusian Г is usually considered to be voiced velar fricative [ɣ].

     Exact nature of Ukrainian Г is debated. Some claim it is voiced glottal fricative [ɦ], while others — voiced pharyngeal fricative [ʕ] (both sample sounds in Wikipedia represent Ukrainian Г badly). Also pronunciation of Ukrainian Г may vary, depending on word etymology and speaker's dialect, sometimes it may turn into Belarusian-like [ɣ]. But most agree that in the "canonical" dialects of Ukrainian language in majority of words (excluding cases of borrowed words when it's sometimes pronounced as [ɣ] just because of mimicking pronunciation of original language, e.g. Greek) it differs from Belarusian Г.

     ^{(However in mid-schools Ukrainian children are usually taught that Ukrainian Г is a voiced counterpart of the Х [x]. Not because it really is (it is not; Belarusian Г [ɣ] is), but rather because no other Ukrainian consonant fits this role better — and such a simplified view is enough for mid-schools.)}

     ^{(Please don't be confused by the "[ɦ] vs. [ʕ]" dilemma. The "[ɦ] vs. [ʕ]" is more theoretical question about right classification of the sound, not about right pronunciation. Per my individual perception, Wikipedia samples for both [ɦ] and [ɣ] represent Ukrainian Г inexactly, and for [ʕ] — even worse (thus Ukrainian Г for me sounds like something between Wikipedia samples for [ɦ] and [ɣ]). Real pronunciation of Ukrainian Г can be heard, for example, in the O. Avramenko's lessons — he begins almost every his lesson with "доброГо дня".)}

   • Do Ukrainian Ґ [Belarusian Ґ] and Russian Г have the same pronunciation?

     In general — yes. It is voiced velar stop [ɡ].

   More info: Ukrainian phonology.

3. There is "alternative sound of Г" in some words of the Russian language (ага, угу, бог, господь). What type of Г is it?

   The "Russian phonology" Wikipedia article says it's either [ɡ] (strict), or [ɣ] (like Belarusian).

   While the "Ge (Cyrillic)" Wikipedia article says it's region-dependent: /ɡ/ generally, /ɣ/ in the Southern Russian dialect and may be /ɦ/ in the regions bordering Belarus and Ukraine.

Partial answer:

2. • Do Ukrainian Г and Belarusian Г have the same pronunciation?

     In general — no.

     Belarusian Г is usually considered to be voiced velar fricative [ɣ].

     Exact nature of Ukrainian Г is debated. Some claim it is voiced glottal fricative [ɦ], while others — voiced pharyngeal fricative [ʕ] (both sample sounds in Wikipedia represent Ukrainian Г badly). Also pronunciation of Ukrainian Г may vary, depending on word etymology and speaker's dialect, sometimes it may turn into Belarusian-like [ɣ]. But most agree that in the "canonical" dialects of Ukrainian language in majority of words (excluding cases of borrowed words when it's sometimes pronounced as [ɣ] just because of mimicking pronunciation of original language, e.g. Greek) it differs from Belarusian Г.

     ^{(However in mid-schools Ukrainian children are usually taught that Ukrainian Г is a voiced counterpart of the Х [x]. Not because it really is (it is not; Belarusian Г [ɣ] is), but rather because no other Ukrainian consonant fits this role better — and such a simplified view is enough for mid-schools.)}

   • Do Ukrainian Ґ [Belarusian Ґ] and Russian Г have the same pronunciation?

     In general — yes. It is voiced velar stop [ɡ].

   More info: Ukrainian phonology.

3. There is "alternative sound of Г" in some words of the Russian language (ага, угу, бог, господь). What type of Г is it?

   The "Russian phonology" Wikipedia article says it's either [ɡ] (strict), or [ɣ] (like Belarusian).

   While the "Ge (Cyrillic)" Wikipedia article says it's region-dependent: /ɡ/ generally, /ɣ/ in the Southern Russian dialect and may be /ɦ/ in the regions bordering Belarus and Ukraine.

Partial answer:

2. • Do Ukrainian Г and Belarusian Г have the same pronunciation?

     In general — no.

     Belarusian Г is usually considered to be voiced velar fricative [ɣ].

     Exact nature of Ukrainian Г is debated. Some claim it is voiced glottal fricative [ɦ], while others — voiced pharyngeal fricative [ʕ] (both sample sounds in Wikipedia represent Ukrainian Г badly). Also pronunciation of Ukrainian Г may vary, depending on word etymology and speaker's dialect, sometimes it may turn into Belarusian-like [ɣ]. But most agree that in the "canonical" dialects of Ukrainian language in majority of words (excluding cases of borrowed words when it's sometimes pronounced as [ɣ] just because of mimicking pronunciation of original language, e.g. Greek) it differs from Belarusian Г.

     ^{(However in mid-schools Ukrainian children are usually taught that Ukrainian Г is a voiced counterpart of the Х [x]. Not because it really is (it is not; Belarusian Г [ɣ] is), but rather because no other Ukrainian consonant fits this role better — and such a simplified view is enough for mid-schools.)}

     ^{(Please don't be confused with the "[ɦ] vs. [ʕ]" dilemma. The "[ɦ] vs. [ʕ]" is more theoretical question about right classification of the sound, not about right pronunciation. Per my individual perception, Wikipedia samples for both [ɦ] and [ɣ] represent Ukrainian Г inexactly, and for [ʕ] — even worse (thus Ukrainian Г for me sounds like something between Wikipedia samples for [ɦ] and [ɣ]). Real pronunciation of Ukrainian Г can be heard, for example, in the O. Avramenko's lessons — he begins almost every his lesson with "доброГо дня".)}

   • Do Ukrainian Ґ [Belarusian Ґ] and Russian Г have the same pronunciation?

     In general — yes. It is voiced velar stop [ɡ].

   More info: Ukrainian phonology.

3. There is "alternative sound of Г" in some words of the Russian language (ага, угу, бог, господь). What type of Г is it?

   The "Russian phonology" Wikipedia article says it's either [ɡ] (strict), or [ɣ] (like Belarusian).

   While the "Ge (Cyrillic)" Wikipedia article says it's region-dependent: /ɡ/ generally, /ɣ/ in the Southern Russian dialect and may be /ɦ/ in the regions bordering Belarus and Ukraine.