Which verbs DO NOT go with 了 and or 过?
过 and 了 are aspect markers, and can be used with any verbs, including 过 and 了 themselves.

过 is the experential aspect, indicating that an event has taken place (or the negation thereof), regardless of timeframe (there is no tense in Chinese). 我去过巴黎二十次。 (I have had the experience of visiting Paris twenty times.) 我没去过越南。 (I have not had the experience of going to Vietnam.) 我从来没看过电影。 (I have never had the experience of seeing a movie.)

了 is the perfectative aspect, indicating that an event has been completed, again regardless of time. 我买了个新脚踏车。 (I have bought a new bike.) 吃了饭以后可以逛逛街。 (We can go for a walk after we have finished eating [future]).

Both aspects can be expressed simultaneously: 我吃过了。 They can also be used with other complemented verbs: 我看得见过了。

Aspects can be used with any verb. The only restriction is context and combining conflicting aspects. For example, using durative 着 with completion aspects 过 and 了 makes no sense.

As for your side question: since there is no tense in Chinese, it does not matter in which time frame these aspects are used. The use of 昨天 or 以前 indicates the time, whereas 过 and 了 indicate completion, regardless of time.