Verbs at the end of a German sentence: When do you use the infinitive form of a verb?
Still a little confused about verbs at the end of the sentence in German. When you use the infinitive form of a verb and when do you use the conjugated form? This goes for questions and statements. What is the rule for putting verbs at the end, besides using modal verbs?
Edit: When do I use the infinitive version, and when do I use a participle/conjugated version, for ich and 2nd and 3rd person?

For example:
I used to live in Kiev
"Früher habe ich in Kiew gelebt."
or
"Früher habe ich in Kiew geleben."

Have you ever lived in Kiev?
"Hast du immer in Kiew geleben?"
"Hast du immer in Kiew gelebt?"

Should I eat ice cream?
"Soll ich Eis essen?" (I think it is 'essen' because I used a modal verb)
"Soll ich Eis esse?"

Question

Verbs at the end of a German sentence: When do you use the infinitive form of a verb?
Still a little confused about verbs at the end of the sentence in German. When you use the infinitive form of a verb and when do you use the conjugated form? This goes for questions and statements. What is the rule for putting verbs at the end, besides using modal verbs?
Edit: When do I use the infinitive version, and when do I use a participle/conjugated version, for ich and 2nd and 3rd person?

For example:
I used to live in Kiev
"Früher habe ich in Kiew gelebt."
or
"Früher habe ich in Kiew geleben."

Have you ever lived in Kiev?
"Hast du immer in Kiew geleben?"
"Hast du immer in Kiew gelebt?"

Should I eat ice cream?
"Soll ich Eis essen?" (I think it is 'essen' because I used a modal verb)
"Soll ich Eis esse?"

Answer

The rule depends on which tense or other verbal construct you use: 
The Perfekt and Plusquamperfekt tenses always use the participle:
- Ich habe ... gemacht. 
- Ich bin ... gegangen.  
- Ich hatte ... gedacht.
With the only (rare) exception that modal verbs in the Perfekt or Plusquamperfekt tense use the infinitive and not the participle: 
- Ich habe ... machen müssen. 
- Ich hätte ... tun sollen. (Konjunktiv mood - a bit more frequent)

Modal verbs as the main verb always take an infinitive: 
- Ich muss ... machen.
- Ich darf ... gehen.

The conjugated form of a verb always goes in the second position in a main clause. In subclauses, it comes at the very end: 
- Ich weiß, dass er ... gemacht hat.
- Das Auto, das an mir vorbei gefahren ist, hat eine schöne Farbe.

PS - a small correction: Your sample sentence "Hast du immer in Kiew gelebt?" translates to "Have you always lived in Kiev?". If you want to express the same as in your English sentence, you should say: "Hast du >jemals

Answer

You are right, you  have to use "essen" because "sollen" is a modal verb. Here are ofther examples with modal verbs:

Kann ich deinen Stift haben?
Ich muss viele Vokabeln lernen.  - 
Ich möchte Kaffee trinken. -In these three examples  , the modal verb gets the inflecion and is on second position

2. Ich habe in Kiew gelebt. -   "Gelebt" is the past participle of the verb leben : 
ich lebe - ich lebte - ich habe GELEBT  > I live - i lived - I have LIVED
When forming the perfect tense, the particple  goes to the end, and the form of "haben" or "sein" is inflected and on second 
position: 
Ich habe viele Vokabeln gelernt. Heute habe ich drei Brötchen gegessen Peter hat Andrea angerufen. Ich bin nach 
Griechenland geflogen. Ihr seid nach Hause gekommen.

3. Subordinate cluaes (Nebensätze) : 
Ich hoffe, dass du bald gesund wirst.  Ich glaube, dass das Wetter besser wird.