Translation:I think this is mine.
A nice guess, but it's really just a grammatical rule. When と acts as a particle indicating something that is thought/said/expressed, the verb preceding と is in plain form, and the final verb determines the politeness of the sentence as a whole. For example: かれはとしょかんに行くといいました -> "He said he's going to the library".
Also what comes before と思います is technically reported speech (in this case your thoughts or speech inside your head). Verbs in reported speech are in plain form.
Yes, present, active (plain form) equivalent to the dictionary form eg. 行く、来る、食べる、分かる、学ぶ of verbs or the verb stem + base 3.
わたしのだ is short for わたしのもの. In this sentence, it is implied that the object is already referred to, therefore repeating it would be redundant (similar to sentences like 赤いのをください) ... (at least I think so)
"I think this is mine", a man says as he is searched after shoplifting a fridge under his jacket.
To show that what proceeds it is 'reported speech' - in other words whatever proceeds と is either speech, or thoughts.
Are you allowed to say と思い in reference to yourself? I always thought you can only use it when "thinking" something about someone else.
There are essentially two sentences in one sentence here これは。。。とおもいます - 'I think ....this thing' and わたしのだ 'it is mine'. わたしのだ is a subordinate clause within the main clause or sentence and it is also reported speech - or in this case 'reported thought". The と before the verb indicates the speaker's thought/s or in other words it lets us know that the subordinate clause that precedes it is reported speech/thought. Verbs in reported speech are in plain form hence we have だ instead of です. Hope this makes sense?